Notes from the Prague Set Theory seminar

Notes from the Prague Set Theory seminar
Jonathan Verner
January 8, 2015
1
Disclaimer
The following are my notes that I am taking at our seminar meetings. I do not claim that they accurately represent
what happens at the seminar. In fact, they probably misrepresent quite a few things. In particular, they have not been
edited in any way and contain many errors. Moreover, no effort has been done to attribute results correctly. None of
the errors should be attributed to the speakers and I bear all responsibility for them. If you spot some error do let me
know so that I can correct it.
The notes are generated from a sourcefile written in a variant of markdown. At the top you should find a pdf version
for print.
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6. 11. 2013
2.1 Definition. Let B be a Boolean algebra and P ∈ Bω a sequence. For a finite A ⊆ ω we define
p(A, P ) =
max {|K| : K ∈ [A]<ω & P [K] is centered}
|A|
and the intersection number of P to be
i(P ) = inf {p(A, P ) : A ∈ [ω]<ω }
The weak intersection number is defined similarly if we consider sequences instead of sets.
2.2 Definition. A filter F is a measure centering filter if for every Boolean algebra B and for every P ∈ Bω with positive
intersection number there is a set A ∈ F such that {p(n) : n ∈ A} is centered.
2.3 Theorem (Foreman). Every rapid filter is a measure centering filter.
2.4 Lemma. If a sequence has a positive intersection number then there are infinitely many centered subsequences.
2.5 Question. What about the weak intersection number?
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3.1
20. 11. 2013
E. Thuemmel: Pure decision property
3.2 Definition. MA has the pure decision property (or Prikry property ), if for each formula ϕ and a condition [s, A] there
is a stronger condition of the form [s, B] deciding ϕ.
3.3 Note. This is equivalent to saying that for each a ∈ RO(MA ) and [s, A] there is a stronger condition of the form
[s, B] such that [s, B] ≤ a or [s, B] ≤ −a.
3.4 Fact (Farah). A coideal A is semiselective iff MA has the pure decision property.
3.5 Example. The coideal corresponding to FIN, an ideal generated by a MAD family and a selective ultrafilter are
selective. However note that the other two are not definable.
3.6 Definition. An ideal I on ω is Ramsey (we also write I + → (I + )22 ) if for each X ∈ I + and each coloring χ :
[X]2 → 2 there is a Y ∈ I + , Y ⊆ X which is monochromatic.
1
3.7 Definition. An ideal is selective if it is Ramsey and P(ω)/I is σ-closed. It is semi-selective if it is Ramsey and P(ω)/I
is ω-distributive.
3.8 Note. We will sometimes say that an ideal or a co-ideal is σ-closed, ω-distributive etc. and will mean, in fact, that
the forcing P(ω)/I is σ-closed etc.
3.9 Example. If we add a Sacks real then groundmodel infinite reals form a semi-selective coideal which is not
selective.
3.10 Observation. selective → semiselective → Ramsey
3.11 Note. If A is left-shift invariant (i.e. A ∈ A → A \ {minA} ∈ A) and MA has the pure decision property then it
adds a dominating real.
3.12 Fact. A semiselective coideal cannot be definable.
3.13 Question. Can we have a definable Ramsey coideal?
3.14 Definition. The ideal conv is the ideal generated by Cauchy sequences of rationals in the interval [0, 1].
3.15 Lemma (2.15, Hruˇsa´ k, Thuemmel & spol.). If an ideal is not ω-distributive and is proper then there is X ∈ I + such
that I X is Katˇetov above conv.
3.16 Question. Can we ommit properness in the above lemma?
3.17 Answer (Thuemmel). No.
3.18 Definition. An ideal has P I iff for all descending sequences hAi : i < ωi of I postive sets such that Ai \ Ai+1 ∈ I
there is a positive pseudointersection.
The following is a combinatorial reformulation of 3.15.
3.19 Proposition. An ideal I + is somewhere Katˇetov above conv iff it is either not (ω, 2)-distributive or not P I .
It will be first instructive to notice the following reformulation of (ω, 2)-distributivity+P I :
3.20 Observation. An ideal I is (ω, 2)-distributivity+P I iff for each matrix {Aji : i < ω, j < i} there is a disjoint ⊆∗ refinement.
Proof. To see this, distributivity gives us a I-disjoint ⊆I refinement and P I allowes to improve this to an almost
disjoint ⊆∗ -refinement).
Proof of Proposition. ” ⇒”. Fix an X ∈ I + and ϕ : X → Q ∩ [0, 1] witnessing that I + is somewhere above conv (i.e. for
all F ∈ conv, ϕ−1 [F ] ∈ I). Let
j j+1
j
−1
Ai = ϕ
,
, i < ω, j < 2i
2i 2i
The Aji s form a matrix. If I is not (ω, < ω) distributive we are done (since then it is also not (ω, 2)-distributive).
f (i)
So assumeSit is. Then there is a function f : ω → ω and a positive set X such that X \ Ai
= Ii is in I. Then
Xn = X \ i<n Ii is a descending sequence of positive sets with small differences. If I had P I then we would have a
f (i)
f (i)
positive Y ⊆ X such that Y ⊆∗ Ai for each i < ω. Then Y ⊆ ϕ−1 [ϕ[Y ]] however ϕ[Y ] is Cauchy (since Y ⊆ Ai
for each i < ω) so it is in conv — a contradiction with Y being positive.
” ⇐”: Assume I is either not (ω, 2)-distributive or not P I . By the above observation (3.20) we may assume that there
is a matrix {Aji : i < ω, j < 2i } which has no almost disjoint refinement. We will define ϕ : ω → [0, 1] as follows:
ϕ(n) =
X i(n)
,
22i+1
i<ω
i(n)
where i(n) is the unique number such that n ∈ Ai . Then we compose ϕ with a suitable function such that its image
is a subset of Q. This will witness that I is Katˇetov above conv.
We will now show the proof of Lemma 3.15.
Proof. Assume I is not ω-distributive and that it is proper. First notice that if I is Proper and (ω, 2)-distributive then
it is ω-distributive. Now apply proposition 3.19.
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Next we turn to give an example showing 3.17.
3.21 Example. Farah has an ideal I which is Ramsey and not semiselective and not ω-distributive. We will improve it
so that it is (ω, 2)-distributive and P I . For u ∈ <ω R we define Au ⊆ ω such that A∅ = ω and {Aua ξ : ξ ∈ R} is a MAD
on Au . Let H consist of all A ⊆ ω which have some Au ⊆∗ A. If we are careful enough when constructing Au this
will be a coideal. (E.g. enumerate P(Au ) as {Bξ : ξ ∈ R} and choose Aua ξ so it is contained in Bξ or its complement).
Now H will not be ω-distributive by construction. If we are even more careful, it will be Ramsey: Assume we have
constructed Au . Enumerate all colorings of [Au ]2 as {χξ : ξ ∈ R} and choose Aua ξ so that it is homogeneous with
respect to χξ . In a similar way we will ensure that H has P I and is (ω, 2)-distributive.
3.22 Note (Balcar). In the above we could have everywhere replaced properness with (ω, ·, ω1 )-distributivity.
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4.1
27. 11. 2013
Wolfgang Wohofsky
Most of the following is part of Wolfgang’s thesis and joint work with Michael Hruˇsa´ k and Ondˇrej Zindulka.
4.2 Definition. A set A ⊆ R has strong measure zero if for each sequence of positive real numbers hεn : n < ωi there is
a sequence of intervals hIn : n < ωi which covers A and such that λ(In ) < εn .
4.3 Note. It is important that we take intervals (or open balls in general Polish spaces) instead of general open sets.
We would just get measure zero with this.
4.4 Note. A strong measure zero cannot contain a perfect set (as it is preserved under uniform continuous images).
4.5 Note. Strong measure zero sets form a σ-ideal.
4.6 Defintion. A set X ⊆ 2ω is meager shiftable if it can be translated away from each meager set, i.e. for each meager
set M ∈ M there is t ∈ R such that X ∩ M + t = ∅.
4.7 Note. A set X is shiftable away from M iff X + M 6= 2ω
4.8 Notation. If I is any collection, let I ∗ be the collection of sets which are shiftable away from each set in I.
4.9 Observation (Pˇr´ıkry).
´ Every meager shiftable has strong measure zero.
4.10 Theorem (Galvin-Mycielski-Solovay). In R the strong measure zero sets are exactly M∗ .
This motivates the following question.
4.11 Question. Given a polish topological group, does the Galvin-Mycielski-Theorem 4.10 hold in it?
The following is a partial answer.
4.12 Theorem. If (G, +) is a locally compact polish group then M∗ (G) = SM Z(G).
The ⊆ inclusion is easy and follows from separability. The hard inclusion is the other one. From this it follows
that under the Borel Conjecture (i.e. there is no uncountable strong measure zero set), the GMS holds for all polish
groups.
Proof. (idea) Say that a group G has the strong GMS property if for each meager set M and there is a sequence
hεn : n < ωi of positive reals such that for each sequence hUn : n < ωi such that d(Un ) < εn there is a t ∈ G such that


\ [

Um  + t ∩ M = ∅
n<ω m≥n
One now shows that a compact group has the strong GMS property. Suppose now that X is strong measure zero and
M is meager. Apply the strong GMS property to M to find the sequence ε = hεn : n < ωi . Apply strong-measure
zero to X and this sequence to get a sequence hUn : n < ωi of open balls with diameters bounded by ε. WLOG we
may assume that


\ [
X⊆
Um 
n<ω m≥n
Now...
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4.13 Example. Under CH there is an uncountable meager-shiftable set in Zω and a SMZ set which is not meagershiftable, i.e.
[Zω ]ω $ M∗ (Zω ) $ SM Z(Zω )
The second part does not need CH.
Proof. The first $ is shown by simple induction. Works for a general ideal J if cov(J ) = cof (J ) = κ. Then there is a
set X of size κ which is in J ∗ . For the second, let
[
sa 0| ·{z
· · 0},
T = Z<ω \
s∈Z<ω
m(s)
where m(s) is the maximum of the range of s or 2. Then [T ] is a counterexample to the strong GMS property for
Zω .
4.14 Definition. An ideal J of subsets of some group G is κ-translatable if for each M ∈ J there is a (possibly larger)
M 0 ∈ J such that for each T ∈ [G]<≤κ there is t ∈ G such that M + T ⊆ M 0 + t.
Notice that if an ideal is κ-translatable then it is a κ+ -ideal.
4.15 Definition (Pˇr´ıkry).
´ We say that a set is strongly meager if it is in N ∗ , i.e. shiftable away from measure null sets.
The Dual Borel Conjecture says that there are no uncountable strongly meager sets. Bartoszynski showed that, in
ZFC, N (measure zero sets) are not 2-translatable, in particular, not ω-translatable.
4.16 Theorem. The meager-shiftable ideal on a locally compact Polish group is ω-translatable.
4.17 Note. It might be that [T ] from the proof of 4.13 is a counterexample to the ω-translatability of M∗ (Zω ).
4.18 Lemma. If (G, +, 0) is a compact group and Θ an open cover of G there is a (Lebesgue neighbourhood) U of 0 such
that for each x ∈ G there is O ∈ Θ such that U + x ⊆ O.
4.19 Note. The diameter of the Lebesgue neighbourhood is the Lebesgue number of the open cover.
4.20
Honza Stary´ / BB
4.21 Definition. A submeasure µ on a Boolean algebra B is exhaustive if for each countable antichain {an : n < ω} ⊆ B
and each ε > 0 the set {n : µ(an ) > ε} is finite.
4.22 Theorem (Kalton,Roberts). Let µ be a uniformly exhaustive submeasure on a Boolean algebra. Then there is a finitely
additive measure m ≤ µ having the same null sets, i.e. N (m) = N (µ).
4.23 Proposition. Let B ≤ C be Boolean algebras and mu an exhaustive submeasure on B. Then µ can be extended to an
exhaustive submeasure on C. If µ was a measure, this extension can also be chosen to be a measure.
Proof. (Basically, an application of the Sikorski theorem) Consider N (µ) and the quotient algebra B/N (µ). Then µ is
strictly positive exhaustive submeasure on the quotient and induces a metric d on the quotient algebra:
d([a], [b]) = µ(a4b)
Let (M, ρ) be the metric completion of (B/N (µ), d). This completion is a complete Boolean algebra. By the Sikorski
extension theorem (i.e. complete B.A.s are injective), we can extend the mapping from B into M to a mapping h :
C → M. Now define
µ(c) = ρ(h(c), 0)
Then µ is the required extension. We will show that it is exhaustive. Let {an : n < ω} be an antichain in C and fix an
ε > 0. By induction find hb0n : n < ωi ⊆ B such that ρ(bn , an ) < ε/2n . Let bn = b0n − b00 ∨ · · · b0n−1 .
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5.1
4. 12. 2013
Honza Greb´ık: Asymptotic Density in Generic Extensions
5.2 Proposition. Let s = han : n < ωi be a sequence of numbers such that for each i 6= j the greatest common divisor
gcd(ai , aj ) is equal to one. Then the set
As = {k ∈ N : (∀n < ω)(an 6 | k)}
4
has density
d(A) =
Y
1
1−
.
an
i<ω
5.3 Example. Given a finite sequence σ of digits {0, . . . , d} the set Aσ of numbers which start with σ when written in
base d does not have density.
5.4 Definition. For k < ω define the set of exponent-k-free numbers
Pk = Ahpk :p is primei = {n : (∀p)(pk 6 | n)}
5.5 Proposition. d(Pk ) = 1/ζ(k), where xi is the Riemann zeta function, i.e.
−1
∞
Y
X
1
1
=
1− x
ζ(x) =
nx
p
n=1
p∈P
5.6 Definition.
(
P2i
=
k:k=
m
Y
)
pi & m = i mod 2
i=1
5.7 Proposition.
1. d(P21 ) = 3/π 2
2. π(x) ∼ x/ log x
3. ζ(x) = 0 → Re(x) 6= 1
5.8
Balcar: Torturing Honza Stary´
Recall ...
5.9 Definition. Let (X, ρ) be a metric space and O an open cover of this space. Then the lebesgue number of O is
defined as
l(O) = sup{ε > 0 : (∀x ∈ X)(∃O ∈ O)(B(x, ε) ⊆ O)}
And now for something entirely different...
5.10 Definition. A T ⊆ B is a tree in the Boolean algebra B if it is a tree in the (inverted) Boolean ordering such that
disjointness in T implies disjointness in B.
5.11 Proposition. Let B be a Boolean algebra, f a strictly positive exhaustive functional on B (i.e. f (0) = 0) which is
nondecreasing. Then each tree T ⊆ B is countable.
5.12 Note. In case f is a measure, this is due to/can be found in S. Koppelberg (CMUC).
Proof. First, since B is ccc (by exhausitivty), each level and branch is countable. Assume, aiming towards a contradiction, T is uncountable. Then ht(T ) ≥ ω1 , i.e. Tα 6= 0 for uncountably many αs. Let rα = max{f (x) : x ∈ Tα }.
This maximum exists by exhaustivity. The following claim will finish the proof, since there is no uncountable strictly
decreasing sequence of real numbers.
5.13 Claim. The sequence hrα : α < ht(T )i contains a strictly decreasing cofinal sequence.
Proof. Note that the sequence is decreasing. It is sufficient to show that for each α the set {β : rα = rβ } is countable.
This follows from the fact that f is exhaustive, that the witnessing elements from the Tβ s cannot form a chain and
¨
from Erdos-Dushnik-Miller.
5.14 Note (E. Thuemmel). This is equivalent to the fact that there is no strictly positive, nondecreasing exhaustive
functional on the Suslin tree. The fact that Suslin does not carry a measure is folklore.
5.15 Problem (Prikry). Does every ccc forcing add a random real or a Cohen real?
5.16 Note. If the measure algebra cannot be embedded into the Talagrand algebra, it might be a candidate for a ZFC
answer to the above problem.
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5.17 Proposition.
Q Let (Bn , µn ) be a sequence of Boolean algebras with finitely additive measures on them and U a free ultrafilter
on ω. . For x ∈ Bn define
µU (x) = U − lim µn (x(n))
Q
Q
Then µU is a finitely additive measure on Bn . Moreover the [0]U ⊆ Q
N ull(µU ) = N ull, hence Bn /N ull(µU ) is a quotient
aglebra of the ultraproduct. Since the ultraproduct is complete then Bn /N ull, as a quotient algebra, has the CSP. By the
Smith-Tarski theorem it must be complete (since it also carries as strictly positive finitely additive measure). We shall show that
µU is in fact σ-additive.
Proof.
We will show that
So let bk be a sequence of elements of BU =
Q
V it is continuous (and hence σ-additive.
Q
B/N ull. such that bk = 0. We may assume that bk ∈
B and that they are strictly decreasing. Assume,
aiming for a contradiction, that
lim µU (bk ) = ε > 0
Then
Uk = n < ω \ k : µn (bk (n)) > ε/2 ∈ U
T
Q
k
k
Then Uk = ∅. We define b ∈
V k Bn as follows. For n ∈ Uk \Uk+1 let b(n) = b (n). Then b ≤ b and, since µU (b) > ε/2,
0 < b. A contradiction with b = 0.
5.18 Example.
5.19 Example. If Bn is the Cohen algebra with a measure extending the standard measure on clopen sets, then BU
is the measure algebra of length c. Proof: This follows, by Fremlin, from the fact that (BU , ρ) has open-hereditary
density c. To show this, consider in Bn a measure1/2-independent system (i.e. µ(xk ) = µ(xk 4xn ) = 1/2 = 1/2µ(xk ∧
xn ), hxk : k < ωi. This gives us c-many pairwise different hbα
n : n < ωi such that
β
ρ(bα , bβ ) = U − lim ρ(bα
n , bn ) = U − lim 1/2 = 1/2.
Hence hd((BU , ρ)) ≥ c.
and the story continues...
5.20
Balcar: Torturing Honza Greb´ık
5.21 Theorem. P(N)/Z0 ' P(N)/f in ∗ B(c).
Proof. Let In = [2n , 2n+1 ). Given X ⊆ ω let
"
f ([X]f in ) =
#
[
n∈X
In
Z0
Then, since
X ∩ [2n , 2n+1 )
= 0,
X ∈ Z0 ⇐⇒ lim
2n
Q
f is a regular embedding. Now consider P(N)/f in ' P(In ) and use the ultrapower technique (5.17).
5.22 Theorem. Let F be a rapid filter. Then
X ∩ [n2 , (n + 1)2 )
G = X : (∃F ∈ F) lim
=1
n∈F
2n + 1
is a filter which cannot be extended to a rapid but, after adding a random real, can.
6
11. 12. 2013
6.1
E. Thuemmel: Exponent
6.1.1
Pure decision
Recall the following question which motivates the subsequent.
6.2 Question. Does pure decision imply that the respective forcing adds a dominating number?
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However, this question is not well defined since there is no general definition of pure decision. As an example of
pure decision consider the following classical theorem:
6.3 Proposition. Mathias forcing M has the pure decision property (Prikry property), i.e. for each sentence ϕ and condition
[s, A] there is a stronger condition [s, B] which decides ϕ.
Our aim is to show that, in a suitable sense, pure decision in fact does impliy that the forcing adds a dominating
number.
We now aim to introduce the exponent. The following is based on E. Thuemmel’s paper Open mappings on extremally
disconnected compact spaces published in AUC Math. et Ph., Vol 47 (2006), No. 2, 73-105.
6.4 Notation. Let us identify each condition [s, A] in the Mathias forcing with the set
[s, A] = {B : s v B &B ∈ [A]ω } ,
thus we can consider each [s, A] to be a (Laver) tree. For each s ∈ [ω]<ω \ ∅ define sh(s) = s \ min s. Consider
M0 = {[s, A] ∈ M : s 6= ∅} and extend sh to a map sh : M0 → M as follows:
sh([s, A]) = [sh(s), A].
This map has the following property
6.5 Observation. For each [s, A] ∈ M0 and [t, B] ∈ M such that [t, B] ≤ sh([s, A]) there is [r, C] ≤ [s, A] such that
sh([r, C]) ≤ [t, B].
Proof. Let r = s ∪ t, C = B.
We shall generalize this observation into a definition:
6.6 Definition. A monotone map ρ : P → Q has property (*) if
(∀p ∈ P )(∀q ≤ ρ(p))(∃r ≤ p)(ρ(r) ≤ q)
6.7 Proposition. If a monotone map ρ : P → Q has property (*) we can extend it to a map ρ∗ : RO(Q) → RO(P ):
ρ∗ (I) = ρ−1 (I)
which is a complete homomorphism.
6.8 Note. If ρ[P ] is dense in Q then ρ∗ is a complete embedding. However, in our setting, the image of P will not be
dense.
We will now consider the extension of the sh map sh∗ : RO(M) → RO(M). This is a complete homomorphism.
Consider the dual space X = St(RO(M)), which is an extremally disconnected compact space, and the continuous
function f = st(sh∗ ) : X → X, which is an open mapping. Given a condition [s, A] ∈ B we will write Us,A for the
basic clopen set given by this condition.
6.9 Observation.
1. f [Us,A ] = Ush(s),A
2. f [U∅,A ] ⊆ U∅,A
6.10 Observation. The set of fixed points of f is nowhere dense.
Proof. {Us,A : [s, A] ∈ M0 } forms a π-base for the topology. Since f [Us,A ] ∩ Us,A = ∅ for each [s, A] ∈ M0 , Us,A contains
no fixed point for each element of the basis.
6.11 Observation. The map f has a fixed point.
Proof. Aiming for a contradiction, assume that no x ∈ X is a fixed point. So we can separate x from f (x) by an open
neighbourhood. By compactness we can find finitely many Ui covering X such that f [Ui ] ∩ Ui = ∅. Since it is a cover,
we can take an arbitrary A and find i such that U∅,A ⊆ Ui . However this is impossible by 2. of observation 6.9.
To summarize, starting from Mathias forcing, we have arrived to an extremally disconnected compact space X with
an open map f : X → X having a nowhere dense set of fixed points. It is surprising that we can, in some sense,
reverse this process. This is the content of the following definition and theorem.
7
6.12 Note. All of this works in the more general setting with Mathias forcing replaced by MA where A is a selective
coideal, or by Prikry forcing.
6.13 Definition. Given a regular cardinal κ, a complete Boolean algebra B and a mapping r : B κ → B which is
1. a κ-complete homomorphism
2. uniform (i..e. if |{α < κ : bα 6= 0}| < κ then r(b) = 0).
3. a retract, i.e. r(cb ) = b for each b ∈ B, where cb is the constant function having value b
Then we define
n
o
<ω
Expκ (B, r) = T ∈ [κ] B : (∀s ∈ [κ]<ω )r(hT (sa {α}) : α < κi) = T (s)
<ω
Then Expκ (B, r) is a complete, κ-complete subalgebra of B [κ]
the tree Tb such that Tb (s) = b.
which completely embeds B by assingning to b ∈ B
6.14 Note. κ must either be ω or, at least, strongly inaccessible. It is conjectured that, if uncountable, it must be
weakly compact and this is also sufficient.
6.15 Example. Consider the characteristic function χ : P(κ) → B κ and r : B κ → B. The composition gives the map
ϕ. If B is kappa-distributive, then each such ϕ determines r:
o
_n
b : ϕ({α : b ≤ bα }) ≥ b
r(hbα : α < κi) =
If M is the Mathias forcing, then let B = Compl((P )(ω)/f in) and ϕr (A) = [A]f in . The previous note gives us rϕ .
Then Expω (B, rϕ ) = RO(M). The regular embedding showing this is defined as follows
[s, A] 7→ T ∈ Expω (B, rϕ ),
where
T (t) =
[A] s v t & t \ s ⊆ A
0
otherwise
6.16 Note. Exponent has fusion in the following sense. Given T1 , T2 ∈ Expκ (B, r) and n < ω we define
T1 ≤n T2 ⇐⇒ (∀s, |s| ≤ n)(T1 (s) = T2 (s))
V
If hTn : n < ωi with T0 (∅) 6= 0 is a fusion sequence, then n<ω Tn 6= 0.
6.17 Note. Exponent has pure decision in the following sense. Given a sentence ϕ a nonzero T ∈ Expκ (B, r) and s
with T (s) 6= 0 then there is a stronger T 0 with T 0 (s) 6= 0 which decides ϕ.
6.18 Note. Given a general exponent Expκ (B, r) we define sh∗ : Expκ (B, r) → Expκ (B, r):
T (sh(s)) s 6= ∅
∗
sh (T )(s) =
T (∅)
s=∅
Similarly as in the case of Mathias forcing, we continue to get an extremally disconnected compact space and an
open mapping with a nowhere dense set of fixed points. That the set of fixed points is nonempty follows from pure
decision, in general, F ix(f ) = St(B).
6.19 Theorem. If X is extremally disconnected compact space, f : X → X an open mapping with a nowhere dense set of fixed
points, then there are κ ∈ Ord, a complete Boolean algebra B, a retract r and an f -invariant clopen subset X 0 ⊆ X such that
Expκ (B, r) can be regularly embedded into RO(X 0 ).
From now we will consider only κ = ω.
6.20 Proposition. Expω (B, r) adds a dominating real.
Proof. Given s ∈ [ω]<ω define
Ts (t) =
1 svt
0 otherwise
Then {Ts : |s| = n} is a partition of 1. If G is a generic filter on Expω (B, r) then for each n < ω thereSis exactly one
s ∈ [ω]n such that Ts ∈ G. Moreover if Ts , Tt ∈ G then s, t must be compatible. So we can let fG = {s : Ts ∈ G}.
Then fG is a dominating function from ω to ω. To show this, we consider an arbitrary g ∈ ω ω ∩ V , a condition T and
s with T (s) 6= 0. Define
1 s v t & (∀|s| ≤ i < |t|)(g(i) ≤ t(i))
Tg,s (t) =
0 otherwise
Now let T 0 = T ∧ Tg,s . Then T 0 ≤ T and T 0 (∀i > |s|)(g(i) ≤ fG˙ (i)).
8
Now if a forcing has pure decision we will, as in the case of Mathias, define a shift and then get to the situation where
we can apply theorem 6.19 and, afterwards, theorem 6.20.
6.21 Example (Matet Forcing). Matet forcing consists of pairs [s, A] where s ∈ [ω]<ω and A is a block sequence and the
ordering is given by [s, A] ≤ [t, B] such that t v s, A is a ”condensation” of B and t \ s is a union of blocks from B.
Then Matet forcing has pure decision. If we define
sh([s, A]) = [sh(s), A]
and proceed as before we would like to show that F ix(f ) 6= ∅. However, this will fail. And, indeed, Matet forcing
does not add a dominating real.
´
The following can be found in the dissertationdissertation of Luz M. Garc´ıa Avila.
´
6.22 Definition (Garc´ıa Avila).
Let PF IN consist of pairs [s, A] where s is a finite block sequence and A is an infinite
block sequence.
´
6.23 Theorem (Garc´ıa Avila).
PF IN has pure decision.
In this case, the shift function sh which drops the first block of s , works nicely and makes it possible to apply our
machinery. However, this is not surprising, as the following theorem shows:
´
6.24 Theorem (Garc´ıa Avila).
The set of minimal points of the generic block sequence is a Mathias real.
7
17. 12. 2013 M. Hruˇsa´ k: Questions & stuff I’ve been working on
7.1
Groups
7.2 Problem (Malychin). Is there a separable Fr´echet topological group which is not metrizable.
7.3 Answer (Hruˇsa´ k, Ariet). Consistently no.
(see Hruˇsa´ k, Ariet: Malykhin’s Problem, preprint)
7.4 Question. Is it consistent that there is a countable Fr´echet group of weight ω2 while there is none of weight ω1 .
7.5 Definition. A γ-set is a set of reals where every ω-cover has a γ-subcover. An infinite open cover U of a space X
is an ω-cover if for every finite subset F ⊆ X there is U ∈ U such that F ⊆ U . It is a γ-cover if each x ∈ X is in all but
finitely many U ∈ U.
7.6 Question. Is it consistent that there is a γ-set of size ω2 and no γ-set of size ω1
7.7 Conjecture. If G is an extremally disconnected group and h : G → 2ω is continuous then there is nonempty open
U ⊆ G such that h[U ] is nowhere dense.
7.8 Note.
1. A yes answer to the conjecture implies that there are no countable e.d. topological groups.
2. A yes answer is equivalent to saying that RO(G) does not add Cohen reals for any e.d. group.
3. If h is a group homomorphism, then the answer is yes.
7.9 Question (Group version of Yefimoff,vanDouwen?). Is there a countably compact topological group with no
convergent sequences.
I have a reformulation of the above. Basically, if one tries to answer the above in a naive way by induction, one
arrives at the following invariant. If it is equal to c, then the induction works.
7.10 Definition.
µ = min κ : (∃A ⊆ [2κ ]ω )(∀A ∈ A, 0 ∈ A)(∃C ∈ [2ω ]ω , C → 0)(∀ϕ : 2κ → 2, group homomorphism)(ϕ ”splits”C ⇒ (∃A ∈ A)(0
7.11 Question. Is µ < c consistent?
9
7.12
Countable Dense Homogeneity
7.13 Definition. Two countable sets A, B are of the same type if there is a homeomorphism h such that h[A] = B. A
space is CDH if there is only one type of dense sets.
A typical example of a Polish space which has two types is the half-open interval. It turns out that, basically, this is
the only possibility for finitely many types, i.e. all examples are of the form X ∪ F where X is CDH and F finite. The
uncountable case is different and motivates the following question
7.14 Question. Is there a Polish space with ℵ1 -many types of countable dense sets?
This is related to the Topological Vaughts conjecture. The following theorem can be found in the paper R. HernandezGutierrez, M. Hruˇsa´ k, J. van Mill: Countable dense homogeneity and lambda setsCountable dense homogeneity and
lambda sets, preprint.
7.15 Theorem (vMill, Hruˇsa´ k, Hernandez). If there is a λ-set of size κ then there is X ⊆ R of size κ which is CDH.
7.16 Question. Is it consistent that there is a Baire CDH set of reals of size < c.
7.17
AD Families
The following is a recent result published in the paper Hruˇsa´ k, Guzm´an: n-Luzin gaps, Top. Appl 160 (2013).
7.18 Definition. A system hAα , Bα i is a Luzin gap if
1. Aα ∩ Bα = ∅
2. Aα ∩ Bβ =∗ ∅
3. (∀α 6= β)(Aα ∩ Bβ ∪ Aβ ∩ Bα 6= ∅)
7.19 Theorem (Hruˇsa´ k, Osvaldo). Assume PFA. If A is AD then ψ(A) is normal iff |A| ≤ ω1 and A does not contain a
Luzin gap.
7.20 Question. Is MA enough in the above theorem?
7.21
Cardinal invariants connected to AD families
7.22 Definition. as is the minimal size of a maximal family of AD subsets of ω 2 containing all vertical lines and
otherwise only partial selectors.
7.23 Proposition. a, non(M) ≤ as
7.24 Theorem (Brendle). It is consistent that a = ω1 , non(M) = ω2 , as = ω3
7.25 Theorem (Hruˇsa´ k & students). If c ≤ ω2 then
as = max{a, non(M)}
.
7.26 Notation. If I is an ideal on ω, let
a(I) = min{|A| : A ⊆ I is AD
7.27 Theorem. If I is tall then
a(I) = min{|A| : A ⊆ I and A is MAD}
7.28 Definition. Given a tall ideal I then
cov ∗ (I) = min{|J | : J ⊆ I & (∀X ∈ [ω]ω )(∃I ∈ I)(|I ∩ X| = ω)}
7.29 Observation. a, cov ∗ (I) ≤ a(I)
The above theorem 7.27 follows from the following by choosing I = ED.
7.30 Theorem. If c ≤ ω2 and I is tall then a(I) = max{a, cov ∗ (I)}.
10
S
Proof. Let {Jα : α < ω1 } be a witness of cov ∗ (I). We will construct a MAD A ⊆ I, where A = α<ω1 Aα . Let A0
be an arbitrary MAD family on J0 of size ω1 . At step α < ω1 consider K = {Jβ ∩ Jα } on Jα . Then K ⊥ (on Jα ) is
countably generated so we can disjointify it and consider it as columns in Jα ' ω 2 . Then extend these columns to a
MAD family Aα of size ω1 . Then these Aα s will have the following properties:
1. Aα is an AD on Jα
S
2. β≤α Aβ is AD
S
3. β≤α Aβ Jα is MAD on Jα .
S
4. β≤α Aβ has size ω1 .
7.31
Generic existence of MAD families
7.32 Definition. A MAD family having property P exists generically if any AD family of size < c can be extended to
a MAD family with property P .
7.33 Observation.
1. a = c is equivalent to the fact that completely separable MAD families exist generically.
2. b = c iff Cohen indestructible families exist generically.
3. In known models Sacks indestructible families exist (either because a < c or because they exist generically due to a cardinal
invariant).
7.34 Question. Do Sacks indestructible MAD families exist (in ZFC).
7.35
Fsigma-ideals
7.36 Question. If I is Fσ is it true that I is Tukey equivalent to [c]<ω (i.e. Top) or I is Tukey below the summable
ideal.
7.37 Question. Does every tall Borel ideal contain a tall Fσ subideal?
7.38 Definition. An ideal I is K-uniform if for every X ∈ I + the ideal I X ≤K I (note that it is always Katˇetov
above)
7.39 Question. Is EDf in (i.e. the ideal generated by graphs of functions under the diagonal) the only tall Fσ ideal
which is K-uniform.
7.40 Question. Is there a tall Borel (Analytic) ideal I such that I + → (I + )22 .
7.41 Note. The above fails for Fσ ideals. However, there is an Fσ ideal J such that ω → (J + )22 and there is a
co-analytic ideal for which it holds.
7.42
Strong Measure Zero
Recall definition 4.2.
7.43 Theorem (Prikry). If X ⊆ R is meager-shiftable (see definition 4.6) then it is strong measure zero.
The same proof works for any separable polish group with left-invariant metric. He then asked about the reverse
implication. A yes answer (on the reals) is due to Galvin-Mycielski-Solovay.
7.44 Question. For which groups does the (Prikry-)Galvin-Mycielski-Solovay theorem hold?
From now on G will always be a Polish group with a left-invariant metric and X will be a separable metric space.
7.45 Definition. A set M ⊆ X is uniformly nowhere dense if for every ε > 0 there is a δ > 0 such that for each point
x ∈ X there is a point y ∈ X such that the ball B(x, ε) \ M centered at x with diameter ε contains B(y, δ). It is
uniformly meager if its covered by countably many uniformly nowhere dense sets. We write UM for the collection of
all uniformly meager sets.
11
7.46 Definition. A group is CLI if it carries a complete left-invariant metric.
7.47 Note. Each Polish group carries a complete metric as well as a left-invariant metric. However there are groups,
e.g. Sω , which are not CLI.
7.48 Theorem. If G is CLI and X ⊆ G has strong measure zero, then it is uniformly meager-shiftable (i.e. X + M 6= G for
each uniformly meager M ).
7.49 Observation. If G is CLI and locally compact then meager sets are uniformly meager.
7.50 Question. For which spaces/groups does meager imply uniformly meager. It should be exactly the locally
compact but there is no proof yet.
7.51 Observation. cpt ⇒ M = UM ⇒ GMS
7.52 Definition. A group has the strong GMS property if for every nowhere dense N there is a sequence hεn : n < ωi
such that for every hUn : n < ωi with diam Un ≤ εn there is g ∈ G such that
!
[
g+
Un ∩ N = ∅
n<ω
It has the weak GMS property if the above is not dense in N .
7.53 Observation. strong GMS ⇒ GMS ⇒ weak GMS.
7.54 Note. The strong and weak GMS properties are absolute properties (i.e. with respect to σ-closed extensions.
7.55 Example. Zω is not weakly GMS.
7.56 Definition. A group G has the Bergman property if every compatible left-invariant metric is bounded.
7.57 Theorem. Compact ⇒ strong GMS ⇒ Bergman.
Proof of the second implication. Suppose G is not Bergman and fix d a compatible unbounded left-invariant metric and
a sequence hWn : n < ωi of disjoint balls whose union is dense and diam Wn → 0 and let N = {xn : n < ω} be the
centers of the balls. And now work a little bit.
7.58 Definition (Hruˇsa´ k). A group G is elastic if for each nonempty open U there is a compatible left-invariant metric
d on G such that diamd U = ∞.
7.59 Obsservation. Bergman ⇒ non-elastic.
7.60 Conjecture. GMS implies non-elastic.
So we have the following diagram:
compact
⇓
loc. compact ⇒
8
⇒
M = UM ⇒
strong GMS ⇒
⇓
GMS
?
⇓
weak GMS
Bergman
⇓
non-elastic
8. 1. 2014
8.1 Lemma. Let D ⊆ ω ω be a countable dense set which is partitioned into countably many finite Dk s so that any infinite
union of the Dk s is dense. Then there is n0 such that for each choice {Lk ∈ [Dk ]≤n0 : k ∈ ω} there is x ∈ ω ω such that
S
S
x ∈ k Lk \ k Lk
8.2 Proposition (Gryzlov). There is a dense D ⊆ ω c of size ω which can be partitioned into countably many finite sets Dk
such that
1. (∀M ∈ [ω]ω )(
S
k∈M
Dk is dense)
2. For each n0 ∈ ω and each choice Lk ∈ [Dk ]≤n0 , k ∈ ω the set
12
S
k∈ω
Lk is closed discrete.
8.3
E. Thuemmel: Exponent (cont.)
8.4 Question. Does each tall Borel ideal contain an Fσ (tall) ideal?
8.5 Theorem. Let I be a tall analytic ideal on ω which is ω-distributive (i.e. such that P(ω)/I is ω-distributive). Then there is
A ∈ I + and a tall Fσ ideal J on A such that J ⊆ I.
We shall first show a proof by Hruˇsa´ k. We will need some notation.
8.6 Notation. Given a set A and its partition A = hAn : n < ωi we let
ED(A) = {B ⊆ A : (∃n0 )(∀i > n0 )(|Ai \ Ai+1 | ≤ n0 )}
Proof. There are two cases. Either there is an I-positive set A and its partition A = hAn : n < ωi such that ED(A) ⊆ I.
Then we are done.
Otherwise force with I + and consider the generic ultrafilter G. Since the first case failed, this ultrafilter is selective.
However, by Theorem 3.3 of M. Hruˇsa´ k, J. Verner: Adding ultrafilters by definable quotients, Rend. Circ. Math. Palermo
60(3) 2011, I must be Fr´echet, so is not tall — a contradiction.
Let us now show a proof via exponents.
8.7 Definition. An ideal I on ω has property SEL if for each hRn : n < ωi partition of ω into sets in I there is an
I-positive selector. It is Ramsey if I + → (I + )22 .
For the definition of selectivity and semiselectivity see the picturepicture or take the following lemma as a definition.
8.8 Lemma. Recall that I is Ramsey iff it has property SEL and is (ω, 2)-distributive. It is semiselective iff it has property SEL
and is ω-distributive and it is selective iff it has SEL and is σ-closed.
In the following we shall consider co-ideals instead of ideals.
8.9 Definition. Given a co-ideal H and A ∈ H a sequence hAi : i ∈ Ai is an H-tower if Ai 4A 6∈ H for each i ∈ A and
Ai ⊆ Aj for i > j. Given such an I-tower A we say that a B ∈ [A]≤ω is a diagonal such that (∀i ∈ B)(B \ (i + 1) ⊆ Ai ).
Let diagω (A be all infinite diagonals of A and diag<ω (A) the set of all finite diagonals.
8.10 Definition. Given a co-ideal H we define
Exp(H) = {[s, A] : s ∈ [ω]<ω & A is an H-tower}
with the extension relation [s, A] ≤ [t, B] defined as follows:
1. t v s, A ⊆ B and Ai ⊆ Bi for all i ∈ A
2. s \ t is a (finite) diagonal of B
or, if we let
[s, A] = {B ∈ [ω]ω : s v B & B ∈ diagω (A)}
then the extension is just inclusion.
If H is distributive then this is a special case of the general exponent (see 6.13), i.e. Exp(H) embeds as a dense subset
of Expω (Compl(P(ω)/I), r), where H = I + and the retract r is defined:
_
r(b) =
[A]H : (∀i ∈ A)([A]H ≤ bi )
.
The embedding takes [s, A] to the tree T where T (t) = [A]H for s v t such that t \ s is a diagonal of A and ∅ otherwise.
Note that classical Ramsey theorems hold also in this setting.
8.11 Theorem (Nash-Williams). For all F ⊆ [ω]<ω there is an infinite A ∈ [ω]ω such that either
1. (∀B ∈ [A]ω )(∃s ∈ F)(s v B) (i.e. F is a barrier) or
2. [A]<ω ∩ F = ∅
13
Mathias generalized this to being able to select A from a selective co-ideal and Farah generalized this further to semiselective co-ideals. In our setting the following holds (basically with the same proof as Farah, only skipping the last
step):
8.12 Theorem (generalized Nash-Williams). If H is an ω-distributive co-ideal then For all F ⊆ [ω]<ω there is an H-tower
A such that either
1. (∀B ∈ diagω (A))(∃s ∈ F)(s v B) or
2. diag<ω (A) ∩ F = ∅
Consider next the Ellentuck theorem.
8.13 Definition. The Elentuck topology on [ω]ω is generated by sets of the form [s, A], where [s, A] = {B ∈ [ω]ω : s v
B ⊆ A}.
Note that a set X has the Baire property in this topology if ∀[s, A])(∃[t, B] ≤ [s, A])([t, B] ⊆ X ∨ [t, B] ∩ X = ∅). A set
is completely Ramsey if ∀[s, A])(∃B ∈ [A]ω )([s, B] ⊆ X ∨ [s, B] ∩ X = ∅)
8.14 Theorem (Ellentuck). The Baire property is equivalent to being completely Ramsey (in [ω]ω with the Ellentuck topology).
Mathias and Farah generalized this taking some selective and semiselective, respectively, ω-distributive co-ideal H
instead of [ω]ω and it also holds in our setting (taking conditions [s, A] ∈ Exp(H)). It is also true that H-Ramsey sets
(R(H)) form a σ-algebra and H-Ramsey null sets (R0 (H)) form a σ-ideal.
8.15 Definition. A Marczewski pair is a pair (B, I) where B is a σ-algebra of subsets of some Polish space Y and I is
a σ-ideal on Y satsifying
(∀X ⊆ Y )(∃Φ(X) ⊇ X, Φ(X) ∈ B)(∀Z)(Z ⊆ Φ(X) \ X & Z ∈ B → Z ∈ I)
8.16 Theorem (Marczewski). If (B, I) is a Marczewski pair then B is closed under the Suslin operation A.
8.17 Theorem (Pawlikowski). Assuming CH the H-Ramsey sets and H-Ramsey null sets form a Marczewski pair.
Using Marczewski’s theorem it follows that, assuming CH, all analytic sets are H-Ramsey. However, this statement
is simple enough, so by some absolutness arguments (due to Platek) the CH assumption may be dropped.
Let us now give an alternative proof of theorem 8.5:
Proof of theorem 8.5. Since I is analytic and ω-distributive we can find an I + -tower A such that either
1. [∅, A] ⊆ I or
2. [∅, A] ∩ I = ∅
Since [∅, A] is a tall, downwards-closed set it generates a tall Fσ -ideal. so in case 1. we are finished. Case 2, however,
easily leads to a contradiction with the tallness of I.
8.18
J. Verner: Generalized Grigorieff forcing
J. Verner gave a (very) short overview of the paper R. Honz´ık, J. Verner: A Lifting Argument for the Generalized Grigorieff
Forcing, to appear in Notre Dame J. of Formal Logic.
The article generalizes standard Grigorieff forcing to uncountable cardinals (see also B. M. Andersen, M. Groszek:
Grigorieff Forcing on Uncountable Cardinals Does Not Add a Generic of Minimal Degree, Notre Dame J. of Formal Logic
50(2) 2009) and shows that it can be used to add unbounded subsets of κ while, e.g., preserving measurability. In the
following we introduce the basic notion and show that, under GCH, it does not collapse cardinals.
8.19 Definition. Given a an uncountable regular cardinal κ and a κ-complete ideal I on κ extending the nonstationary
ideal N S we define the generalized Grigorieff forcing
PI = {f ∈ A 2 : A ∈ I}
which consists of partial characteristic functions of subsets of κ ordered by extension.
8.20 Note.
1. We want the forcing to be κ complete so that it does not add bounded subsets of κ.
14
2. Since we want the ideal to be κ complete, it does not make sense to consider singular cardinals.
3. We want the ideal to extend N S because that allows us to use Fodors lemma in the arguments. The situation
for κ-complete ideals not extending N S clear.
In the situation above (i.e. κ uncountable, regular, I κ-complete, extending N S) we have the following theorem.
8.21 Theorem. Assume GCH and 2<κ < κ. Then PI does not collapse cardinals iff I is closed under diagonal unions.
The diagonal union is the dual notion to the diagonal intersection and is defined as follows:
8.22 Definition. Given a sequence A = hAα : α < κi, its diagonal union is
5A = {β < κ : (∃α < β)(β ∈ Aα )}
Proof. If an ideal is not closed under diagonal unions we first show that it is not a ”generalized P-ideal”. Then we
just copy the standard proof that Grigorieff forcing with a non-P-point collapses the continuum. This direction does
not need the GCH. For the other direction, since I is κ-closed and, under GCH, κ++ -cc we need only show that it
does not collapse κ+ . For this we use fusion.
8.23 Definition. Given two conditions p, q ∈ PI we define
p ≤α q ⇔ p ≤ q & dom(p) ∩ (α + 1) = dom(q) ∩ (α + 1)
A sequence hpα : α < κi is a fusion sequence if it satisfies:
1. pα+1 ≤α pα for all α < κ
S
2. pβ = α<β pα for limit β
It is routine to check that if the ideal I is closed under diagonal unions then it is closed under limits of fusion
sequences. Fusion is now used to show that each function f˙ : κ → κ+ in the extension is bounded. This is done by
constructing a fusion sequence of length κ such that its limit forces that the function is bounded. The crucial step is
the following claim, which uses κ-completeness of I:
8.24 Claim. If p ∈ PI is a condition and α < κ then there is a set A of size 2α (so < κ by assumption) and a condition
q ∈ PI such that q f˙(α) ∈ A and q ≤α p.
8.25 Note. The assumption that 2<κ < κ is in fact superfluous. If κ is a regular limit then it follows from GCH. If not,
we can replace the assumption 2<κ < κ with 3κ at the cost of a more elaborate argument. If κ is a successor 3κ then
follows from GCH. So the only case where the proof breaks down is κ = ω1 where we need to additionally assume
3.
9
15. 1. 2014
9.1
A. A. Gryzlov: On Dense subsets of Tichonoff products (preprint)
9.2 Theorem. Let D be a Tichonoff product of c-many infinite discrete countable spaces. Then there is a countable subset Q ⊆ D
which can be partitioned into finite sets {Qk : k < ω} such that
1. For each infinite C ⊆ ω the set
S
k∈C
Qk is dense in D
2. If F ⊆ Q si bounded in Qk ’s, i.e. if there is n < ω such that for each k < ω the set Qk ∩ F has size at most n, then Q \ F
is dense in D
3. If F ⊆ Q is bounded in Qk ’s (as in 2), then F is closed discrete in Q.
9.3 Note.
1. condition 3. has no chance to hold in ω ω .
2. (E. Thuemmel) condition 3. says that the ideal of sets having discrete closure (in Q) contains EDf in .
Recall the (special case of the) classical theorems
15
9.4 Theorem (Hewitt-Marczewski-Pondycz´ery). The product of c-many separable spaces is separable.
9.5 Theorem (Rasiowa-Sikorski). There is an independent family {Aα,n : α < c, n < ω} of countable partitions of ω into
infinite sets, i.e. such that whenever A ∈ [c]<ω and g : A → ω
\
|
Aα,g(α) | = ω.
α∈A
Proof. Let H = {σ : P(k) → k : k < ω}. It is sufficient to partition H. However, this independent partitioning can be
easily defined as follows For X ⊆ ω let
AX,n = {σ : σ(X ∩ dom(σ)) = n}.
Proof of HMP. To get a dense set of functions f : c → ω, define fσ (X) = σ(X ∩ dom(σ)) for each σ ∈ H.
Proof of the main theorem,3.1. For k ∈ ω let Hk = {σ : σ ∈ P(k) (P(k) \ {∅}), (∀i < k)({i} ∈ rng(σ))}. Note that the Hk s
form a partition of H. For X, Y ⊆ ω and k < ω let
Ak (X, Y ) = {σ ∈ Hk : σ(X ∩ k) = Y ∩ k}
and
[
A(X, Y ) =
Ak (X, Y ).
k<ω
Define now a family M = {A(X, Y ) : X, Y ⊆ ω, Y 6= ∅}}. Notice that for each X ⊆ ω the family MX = {A(X, Y ) :
Y ⊆ ω, Y 6= ∅} is an almost disjoint partition of H. Moreover the family M is independent.
9.6 Lemma (3.1). Let k0 ∈ ω, σ ∈ Hk0 and F ⊆ H such that
1. (∀k < ω)(|F ∩ Hk | ≤ 1)
2. (∀k ≤ k0 )(F ∩ Hk = ∅)
Then for each X ⊆ ω there is Y 6= ∅ such that A(X, Y ) ∩ F = ∅ and σ0 ∈ A(X, Y ).
Proof. Let C = {k ∈ ω : F ∩ Hk 6= ∅} and C0 = C ∪ {k0 }. For k ∈ C define τk such that Hk ∩ F = {τk }. Construct
Y by induction as an increasing union of Yk ’s for k ∈ C. Let σ0 (X ∩ k0 ) = Yk0 . Suppose that k ∈ C and, for k 0 < k,
k 0 ∈ C we have already constructed Yk0 . An easy argument shows how to construct Yk .
9.7 Lemma (3.2). Let σ, τ ∈ H be distinct. Whenever Γ ⊆ P(ω) has size < c then there is X ∈ P(ω) \ Γ and a nonempty
Y 6= ∅ such that σ ∈ A(X, Y ), τ 6∈ A(X, Y ).
Proof. We have σ ∈ Hk(σ) and τ ∈ Hk(τ ) .
Case 1: If k(σ) > k(τ ) then, by definition of Hk , there is X 0 ⊆ k(σ) such that σ(X 0 ) = {k(τ )}. Choose Y such that
Y ∩ k(σ) = {k(τ )}. ...
Case 2: If k(σ) = k(τ ) then, since σ, τ are distinct, there is X0 ⊆ k such that σ(X0 ) 6= τ (X0 ). Since the set {X : X ∩
k = X0 } has size c, there is an X in this set, such that X 6∈ Γ ...
Case 3: If k(σ) < k(τ ) apply lemma 3.1 (9.6).
9.8 Lemma (3.3). There is a set ∆ ⊆ P(ω) of size c and a function π : ∆ → P(ω) \ {∅} such that for σ ∈ H and F ⊆ H
satisfying |F ∩ Hk | ≤ 1 for each k there is a finite ∆(F, σ) ⊆ ∆ for which
\
σ ∈ {A(X, π(X)) : X ∈ ∆(F, σ)}
while at the same time
F \ {σ} ∩
\
{A(X, π(X)) : X ∈ ∆(F, σ)} = ∅.
Proof. The presenter went too fast and used lemma 3.1 ( 9.6) and 3.2 (9.7).
16
Using the lemmas we obtain an independent matrix
M2 = {A(X, Y ) : X ∈ ∆, Y ⊆ P(ω) \ {∅}}.
For each X ∈ ∆ choose an infinite ∇X ⊆ P(ω) \ {∅} countable set such that π(X) ∈ ∇X and
[
{A(X, Y ) : Y ∈ ∇X } = H.
The ∇X gives us a matrix
M3 = {A(X, Y ) : X ∈ ∆, Y ∈ ∇X }
We now disjointify this matrix to a matrix M30 while maintaining A0 (X, π(X)) = A(X, π(X)) and A0 (X, Y ) =∗
A(X, Y ). This matrix is as required (i.e. finish the proof as when prooving HMP from Rasiowa-Sikorski).
9.9
D. Chodounsky:
´ PID and Tukey Maximality
The following is based on the preprint D. Chodounsky,
´ P. Borodulin-Nadzeja: Hausdorff gaps and towers in P(ω)/Fin,
arxiv:1302.4550.
9.10 Definition. Recall that a poset P is Tukey-reducible to Q if there is a cofinal function f : Q → P . A poset P of
size (or cofinality) at most ω1 is said to have Tukey-type [ω1 ]<ω if there is P 0 ∈ [P ]ω1 such that any infinite part of P 0 is
unbounded (in P).
We have the following Tukey-types of posets of size at most ω1 .
1 ≤ ω ⊥ ω1 ≤ ω × ω1 ≤ [ω1 ]<ω
Also recall that
9.11 Definition. PID is the following statement. Whenever I ⊆ [ω1 ]≤ω is an ideal then there are exactly two possibilities:
1. (∃K ∈ [ω1 ]ω1 )([K]≤ω ⊆ I) or
2. ω1 can be partitioned into countably many sets {An : n < ω} such that for each I ∈ I and n < ω I ∩ An = ∅.
9.12 Note. The ideal I consisting of subsets of a Suslin tree which hit every branch below some level only finitely
often (or something like that) does not satisfy either 1. or 2. In particular, PID implies there are no Suslin trees.
Under PFA these five types are the only possible types.
9.13 Theorem (Todorˇcevi´c, Raghavan). Assume PID. Then the existence of precisely 5 Tukey types is equivalent to ω1 < b
+ something.
David and Piotr Borodulin-Nadzeja were interested in Tukey-types of ( ω1 -generated) ideals on ω. They proved the
following theorem.
9.14 Theorem (Chodounsky,
´ Borodulin-Nadzeja). Assume ω1 < b and PID. Then each (nontrivial) ω1 -generated P-ideal
has Tukey type [ω1 ]<ω , i.e. is maximal.
They were, in fact, interested mainly in ideals generated by towers. The following definition is relevant in this context
9.15 Definition. An ( ⊆∗ -increasing) tower T = {Tα : α < ω1 } has property (H) if for all α < ω1 and n < ω the set
{ξ < α : Tξ \ Tα ⊆ n}
is finite.
9.16 Proposition. An ideal generated by a tower T is Tukey maximal iff T has property (H).
Proof. Let T be a tower. We define I ⊆ [ω1 ]≤ω . A set I ∈ I iff the sets Cαn (I) = {ξ ∈ I ∩ α : Tξ \ Tα ⊆ n} are finite
for each α < ω1 and each n < ω. The idea is that we want to construct a subtower satisfying (H) and each I ∈ I is a
potential initial segment of such a sequence, i.e. it has to satisfy (H) and, moreover, we want any such I to always be
extendible to a larger I which is still a potential initial segment. We will show that I is a P-ideal and then apply PID.
In fact, we will show something slightly more general. Consider the (bad) sets
Bαn = {ξ < α : Tα \ Tα ⊆ n}
and B = {Bαn : α < ω1 , n < ω}. Then B ⊥ ∩ [ω]≤ω = I. I.e. the ideal I is the orthogonal of B which has size ω1 . It
immediately follows that it is an ideal. As is typical in PID proofs, we will use the following lemma:
17
9.17 Lemma. If ω1 < b then any orthogonal to a set B of size at most ω1 is a P-ideal.
Proof. Consider any countable sequence hIn : n < ωi ⊆ I as a sequence of vertical lines. Since I is orthogonal to
B each
S B ∈ B gives rise to a function fB : ω → ω. SInce ω1 < b, they are dominated by some f : ω → ω. Then
I = n<ω In \ f (n) is also orthogonal to B.
Now we are ready to apply PID to I. There are two cases to consider. The first condition of PID immediately gives
us the required subtower. It will, therefore, be sufficient to show that the second condition of PID is impossible, i.e.
for each A ∈ [ω1 ]ω1 we need to find an I ∈ I ∩ [A]ω .
9.18 Claim. There is X ∈ 2ω such that X ∈ {Tα : α ∈ A} (closure taken in the Cantor space) and X is not in the ideal
I(T ) generated by the tower T .
Proof. Otherwise {Tα : α ∈ A} is a closed subset of I(T ). Moreover I(T ) is generated from {Tα : α ∈ A} by Boreloperations, i.e. I(T ) is Borel (Analytic will be sufficient for our purposes). However, using Solecki’s characterization
of analytic P-ideals, I(T ) must be generated by at least d many sets — a contradiction.
Consider now {Tα : α ∈ I} for some I ∈ [A]ω . Use the claim to get an x. Suppose I 6∈ I. Then there are β < ω1 , n < ω
such that Tα ⊆ Tβ ∪ n — a contradiction.
9.19 Note. The proof of Todorˇcevi´c’s and Raghavan’s result is very similar however, they need to guarantee that not
only analytic P-ideals are generated by d-many sets but all Fσ -ideals.
10
10.1
22. 1. 2014
Ch. Brech: Application of PID to Banach spaces
(see also the Winter School 2014 talk Ch. Brech: On PID and biorthogonal systems)
10.2 Definition. For a Banach space X a family of pairs hxα , fα : α ∈ Γi ⊆ X × X ∗ is a bi-orthogonal system if
fα (xβ ) = δαβ .
10.3 Note. The span of the first (or second) coordinates need not be the full space (dual space).
Considering
10.4 Fact. Separable infinite dimensional Banach spaces have infinite bi-orthogonal systems. (in fact even satisfying
the stronger condition mentioned in the above note).
it is natural to ask
10.5 Question. Suppose X is a Banach space of density κ. Is it true that it has a bi-orthogonal system of size κ?
10.6 Example (Kunen, ’80). Assuming CH, there is a scattered, compact, Hausdorff, non-metrizable space K of
weight ω1 such that all finite powers are hereditarily separable.
10.7 Example (Todorˇcevi´c, ’89). Assuming b = ω1 there is a space as in 10.6
Both of the above examples yield a nonseparable Banach space with no uncountable bi-orthogonal system, viz X =
C(K). X will be Lindelof in the weak topology and, if hxα , fα : α < ω1 i is a bi-orthogonal system, then {xα : α < ω1 }
will be discrete in the weak topology — a contradiction. (To see this note that the basis of the weak topology consists
of sets of the form
V (x, f0 , . . . , fn , ε) = {y ∈ X : (∀i ≤ n)(|fi (y) − fi (x)| < ε)}
10.8 Theorem (Todorˇcevi´c, 2006). Assuming PID and p > ω1 then every nonseparable Banach space has an uncountable
bi-orthogonal system.
See 9.11 for the definition of the P-ideal dichotomy (PID).
10.9 Question (Todorˇcevi´c). Assume PID. Is b = ω1 equivalent to the existence of a nonseparable Banach space with
no uncountable bi-orthogonal system.
18
10.10 Definition. An Asplund Banach space is a space whose separable spaces have separable duals.
10.11 Fact. If Y is scattered then C(Y ) is Asplund.
10.12 Definition. For a Banach space X a family of pairs hxα , fα : α ∈ Γi ⊆ X × X ∗ is a ε-bi-orthogonal system if
|fα (xβ )| < ε for each α 6= β.
10.13 Theorem (Brech). Assume PID and b > ω1 then every nonseparable Asplund space has an uncountable ε-bi-orthogonal
system for each 0 < ε < 1.
Proof of claim. Hahn-Banach extension theorem ( 14.3)
S means we can extend bi-orthogonal from subspaces. In particular we may assume d(X) = ω1 . Write X = α<ω1 Xα as an increasing union of separable closed subspaces.
By induction construct xα ∈ Xα+1 \ Xα and hα ∈ X ∗ such that hα (xα ) = 1, ||hα || = 1 and hα Xα = 0. Let
S = {hα − hβ : β 6= α}. The proof now splits into three successive claims. First, fix a D ⊆ X dense, Q-linear subspace
of X of size ω1 .
10.14 Claim. There is an uncountable subset {fα : α < ω1 } ⊆ S such that for each x ∈ D the sequence hfα (x) : α <
ω1 i ∈ c0 (ω1 )(= ({t : (∀ε > 0)(|{α < ω1 : t(α) ≥ ε}| < ω)}.
Proof of claim. Apply PID to the ideal
I = {A ∈ [S]ω1 : (∀x ∈ D)(hfα (x) : α ∈ Ai ∈ c0 (A))}
(which is a P-ideal under b > ω1 ) to find the uncountable subset of S.
10.15 Claim. There is an uncountable Γ ∈ [ω1 ]ω1 such that
(∀x ∈ D)(hfα (x) : α ∈ Γi ∈ l1 (Γ)),
where
n
o
X
l1 (Γ) = t ∈ Γ X :
|h(α)| < ∞
α∈Γ
Proof of claim. Let
n
I = A ∈ [ω1 ]ω : (∀x ∈ D)
!
X
|fα (x) < ∞
o
α∈A
First show, using b > ω1 , that I is a P-ideal. Now apply PID to I. The first possibility gives Γ = K so it is sufficient to
show that the second possibility is impossible :-) But that is easy, since any sequence in c0 always contains an infinite
summable subsequence.
The following claim now finishes the proof.
10.16 Claim. There is a sequence hαξ : ξ ∈ ω1 i ⊆ Γ and hxξ : ξ ∈ ω1 i ⊆ X such that hxξ , fξ : ξ ∈ ω1 i is an
ε-bi-orthogonal system.
10.17
J. Lopez-Abad: Families of finite sets
First a remark concerning Christina’s talk.
10.18 Definition. An Auerbach base for a finite dimensional normed space X is a sequence hxi , fi : i < dim(X)i ⊆
X × X ∗ which generatex X and such that fi (xj ) = δij and, crucially, ||fi || = ||xi || = 1 for i < dim(X).
10.19 Fact. Every finite dimensional normed space has an Auerbach basis.
10.20 Theorem (Pelczynski). If X is an infinite dimensional separable Banach space then it has, for each ε > 0 a Markushevich basis (i.e. a sequence hxi , fi : i < ωi ⊆ X × X ∗ , fi (xj ) = δij , such that the first coordinates span a dense subspace of X)
such that 1 − ε < ||fi ||, ||xi || < 1 + ε).
Now we will consider families of finite subsets of S (typically, S = N) and the two relations ⊆, v.
19
10.21 Notation. Given M ∈ [S]ω and a family F ⊆ [S]<ω we let
F M = F ∩ P(M )
(the restriction) and
F[M ] = {s ∩ M : s ∈ F}
(the trace)
Let us consider when F M or F[M ] are ⊆ or a v antichains. When are they ⊆ or v hereditary.
10.22 Observation. The family F is compact iff every sequence in F has a ∆-system subsequence with root in F.
10.23 Definition. The family F is precompact iff every sequence in F has a ∆-system subsequence.
10.24 Note. F is precompact iff its closure (in 2ω ) is compact.
10.25 Example. Given l < ω the family [ω]l is a precompact antichail while its closure, [ω]≤l , is hereditary and
compact.
10.26 Theorem (Ramsey). For any coloring χ : [ω]l → n there is an infinite M ⊆ ω such that χ [M ]l is constant.
10.27 Definition. A family F is Ramsey if for any coloring χ : F → n there is an infinite M such that |{i < n : χ−1 {i}∩
F M }| ≤ 1
10.28 Theorem (Nash-Williams). For a family F the following are equivalent: 1. F is Ramsey 2. there is an infinite M such
that F M is a v-antichain (it is thin) 3. there is an infinite M such that F M is a ⊆-antichain (it is a Sperner system)
10.29 Definition (Nash-Williams). Let F ⊆ [M ]<ω . It is called a barrier on M if it is Sperner and is unavoidable, i.e.
every infinite subset of M has an initial part in F. It is called block on M if it is a thin unavoidable family.
10.30 Proposition. For every thin (Sperner) family F there is an infinite M such that F M = ∅ or F M is a barrier.
10.31 Fact. If F is a barrier on M then for every coloring χ : F → n there is an infinite N ⊆ M such that χ (F N )
is constant.
10.32 Fact. If F is precompact then there is an infinite set such that its trace F[M ] is the closure of a barrier.
10.33 Observation. If B is a barrier, then its topological closure is equal to its ⊆-closure and its v-closure.
In particular, every precompact family has a ⊆-hereditary trace. For example, if F = [ω]n then F[2N] is hereditary.
⊆
10.34 Theorem. Suppose F is arbitrary then there exists an infinite M such that 1.[M ]<ω = F[M ] or 2. F[M ] is the closure
of a barrier on M.
¨
10.35 Theorem (Erdos-Rado).
For each n, l < ω and a coloring χ : [ω]l → n there is I ⊆ l and an infinite M such that for
each s = {n1 < · · · < nl }, t = {m1 < · · · < ml } ∈ [M ]l c(s) = c(t) iff (∀i ∈ I)(ni = mi ).
How do coloring of barriers behave?
¨
10.36 Theorem (Pudl´ak-Rodl).
For each barrier B on M and a coloring χ : B → X there is an infinite M and a barrier C on
ˆ is 1-1 and ϕ(s) ⊆ s.
M, a ϕ : B M → C and a χ
ˆ : C → X such that χ
ˆ ◦ ϕ = χ, chi
10.37 Definition. Let {mn : n < ω} = M be an increasing enumeration. A 0-uniform family on M is {∅}. An α + 1uniform family on M if
F{mn } = {s : mn < s & s ∪ {mn } ∈ F}
is α-uniform in {ml : l > n}. If α is limit, then a family F is α-uniform if there is an increasing sequence αn converging
to α such that F{mn } is αn -uniform.
¨
10.38 Theorem (Pudl´ak-Rodl).
TFAE 1. There is an infinite M such that B M is a barrier on M 2. There is an infinite M
such that B M is uniform on M
10.39 Theorem. If ϕ : B → F in where B is a barrier and ϕ[B] is precompact then there is an infinite M such that for each
s ∈ B M we have ϕ(s) ∩ M ⊆ s.
10.40 Corollary. If ϕ : B → c0 is a mapping of a barrier into c0 whose image is relatively compact in the weak topology then
for each ε > 0 there is an infinite M such that ∀s ∈ B M we have
X
|ϕ(s)|n < ε
n∈M \s
20
10.41 Example. The set S = {s : |s| = min s} is ω-uniform, precompact and large in the following sense.
10.42 Definition. A family F is large in M iff for each N ⊆ M and for all n < ω there is s ∈ F such that |s ∩ N | ≥ n.
It is λ-filling (for λ ∈ [0, 1]) if for each s ⊆ M there is t ∈ F s such that |t| ≥ λ · |s|.
10.43 Theorem. For a cardinal κ the following are equivalent: 1. There is a compact, hereditary, large family on κ. 2. κ is not
ω-Erd¨os. 3. some statement about Banach spaces
¨ iff for any coloring χ : [κ]<ω → 2 there is an infinite set A ⊆ κ such that on
10.44 Definition. A cardinal κ is ω-Erdos
<ω
[A] the coloring χ only depends on the cardinality.
The following is Fremlin’s DU-problem.
10.45 Question. Does there exist a 1/2-filling compact (or precompact) family on ω1 ?
10.46 Note. Taking the Cantor set 2ω instead of ω1 and wanting the family to be Borel (or Analytic?) the answer is
no (see Dodos, P. and Kanellopoulos, K: On filling families of finite subsets of the Cantor set, Math. Proc. Cam. Phil. Soc.
145 (2008), pp 165-175, doi:10.1017/S030500410800109610.1017/S0305004108001096)
10.47
A. Avil´es: Tukey classification of orthogonals
The following theorem can be found in Avil´es, A., Plebanek, G., Rodriguez, J.: Measurability in C(2κ ) and Kunen
cardinals, Israel Journal of Math. 195 (2012), pp 1-30, doi:10.1007/s11856-012-0122-010.1007/s11856-012-0122-0.
10.48 Theorem (Avil´es, Plebanek, Rodriguez). Assume Analytic Determinacy. If I is an analytic family of subsets of ω then
the orthogonal
I ⊥ = {A ⊆ ω : (∀B ∈ I)(|A ∩ B| < ω)}
(which is co-analytic) is Tukey-equivalent (see 9.10) to either 1. {0}, 2.ω, 3. ω ω , 4. K(Q) (i.e. compact subsets of the rationals
ordered by inclusion) or finite subsets 5. R.
10.49 Theorem (Todorˇcevi´c). Let I and J be two orthogonal (i.e. J ⊆ I ⊥ analytic ideals on ω. Then either I and J are
countably separated (i.e. there is a countable C ⊆ P (ω), such that for any disjoint a ∈ I, b ∈ J there is a c ∈ C such that a ⊆ c
and b ⊆ ω \ c). or there is a 1-1 function u : 2<ω → ω such that 0-chains go to elements of I and 1-chains go to elements of J
(where an i-chain is a sequence {x0 , x1 , . . .} ⊆ 2<ω such that for all n we have xn+1 = xa
n (i, k1 , . . . , kpn ) for some k1 , . . . , kpn
).
10.50 Note. In the above theorem, analytic determinacy gives the same result for Σ12 ideals.
Applying the theorem of Todorˇcevi´c to the ideal I and I ⊥ , the second option gives [R]<ω .
10.51 Theorem (Avil´es, Todorˇcevi´c). I, J are countably separated iff there is a metrizable compactification K of ω and a
decomposition K \ ω = U ∪ V such that (∀a ∈ I)(aK ⊆ U ) and (∀b ∈ J )(b
K
⊆V)
This theorem directly reduces 10.48 to Fremlin’s characterization:
10.52 Theorem (Fremlin). If E is a co-analytic metric space then K(E) is Tukey-equivalent to 1,2,3 or 4.
11
11.1
4. 2. 2014
Arturo Antonio Martinez-Celis Rodriguez: Porous Sets
The origin of the following definition goes back to the beginning of the 20th century.
11.2 Definition. Let (X, δ) be a metric space. For a set A ⊆ X define
P (x, R, A) = sup{r ≥ 0 : (∃z)(B(z, r) ⊆ B(x, R) \ A)}
and, for x ∈ X
px,A = lim+
R→0
P (x, R, A)
,
R
px,A = lim+
R→0
P (x, R, A)
.
R
We say that a set A is upper (lower) porous if for every x ∈ X we have px,A > 0 (px,A > 0). The ideals U P (X) and
SP (X) are the σ-ideals generated by, respectively, upper porous and lower porous sets.
21
Recently, the cardinal invariants of these ideals were investigated by several people. The following is a short summary of the results:
The following is from J. Brendle: The additivity of porosity ideals, Proc. Amer. Math. Soc., vol 124 (1) 1996
11.3 Theorem (Brendle, 96). add(U P ) = ω1 , cof (U P ) = c.
The following is from M. Repicky:
´ Cardinal invariants related to porous sets, Set theory of the reals (Ramat Gan, 1991),
433-438, Israel Mathematical Conference Proceedings 6.
11.4 Theorem (Repicky,
´ 91). cov(U P ) ≤ cov(N ), non(U P ) ≥ t, non(U P ) ≥ add(N ).
The following result is from the paper M. Hruˇsa´ k, O. Zindulka: Cardinal Invariants of monotone and porous sets, Journal
of Symbolic Logic, vol 77 (1), 2012
11.5 Theorem (Hruˇsa´ k, Zindulka, 12). Con(non(SP ) < mσ−centered ), Con(cov(SP ) > cov(N )).
We will work with a different definition of the ideal SP which is easier to work with.
11.6 Definition. A subset A ⊆ 2ω is strongly porous if there is an n < ω such that for each s ∈ 2<ω there is a t ∈ 2n
such that [sa t] ∩ A = ∅.
11.7 Lemma (Hruˇsa´ k, Zindulka). SP (2ω ) is the σ-ideal generated by strongly porous sets.
11.8 Theorem (Hruˇsa´ k, Zindulka). The additivity, covering, non and cofinality invariants for SP (R) and SP (2ω ) coincide.
11.9 Note. For Q the cardinal invariants are different!
11.10 Notation. For σ : 2<ω → 2n we let
Xσ = {x ∈ 2ω : (∀k ∈ ω)(x 6∈ [x k a σ(x k)}
11.11 Lemma. Each Xσ is a closed strongly porous set, and each strongly porous set is contained in a set of the form Xσ for
some σ.
11.12 Definition. A forcing P strongly preserves non(SP ) if for each P-name X˙ for an old strongly porous set, i.e.
such that P X˙ ⊆ 2ω ∩ V , then P (∃Y ∈ SP ∩ V )(X˙ ⊆ Y ).
The next lemma gives some examples for strong preservation of non(SP ).
11.13 Lemma. If P is σ-n-linked for every n < ω then P strongly preserves non(SP ).
Proof of Claim. Let ϕ˙ be a name for a function 2<ω → 2n such that
P X˙ ⊆ Xϕ˙ ∩ V.
Write P =
S
n<ω
Pn be a decomposition into 2n -linked subsets. Embed P into its completion B.
11.14 Claim. For every n < ω and for every t ∈ 2<ω there is s ∈ 2n such that for each p ∈ Pn
p ∧ ||ϕ(t)
˙
= s|| =
6 ∅
Proof of Claim.
By the above claim we can fix functions ϕn : 2<ω → 2n witnessing the claim. The following claim now finishes the
proof:
S
11.15 Claim. P Xϕ˙ ∩ V ⊆ n<ω Xϕn
Proof of Claim.
11.16 Lemma. Let A = {B ∈ Borel(2ω ) : µ(B) > 1/2} ordered by inclusion. Then A is σ-n-linked.
Proof. Let n < ω and for every clopen C let
AC = {A ∈ A : µ(C \ A) < 1/n(µ(C) − 1/2)},
and then the blackboard was erased.
22
We will need the following result:
11.17 Theorem (Hruˇsa´ k, Zindulka). Finite support iteration of forcings strongly preserving non(SP ) strongly preserves
non(SP )
11.18 Theorem. It is consistent that non(SP ) < add(N ).
Proof. Start with a model of CH and let P be a f.sp. iteration of length ω2 of the amoeba forcing A. Then V [G] |=
add(N ) = ω2 . By the previous theorem (11.17), P strongly preserves non(SP ). So V [G] |= 2ω ∩ V 6∈ SP so V [G] |=
non(SP ) = ω1 .
11.19 Definition. Let M on be the σ-ideal generated by GO-subspaces of R2 .
11.20 Theorem (Hruˇsa´ k, Zindulka). non(M on) = non(SP ) and cov(M on) = cov(SP ).
The following is also true add(M on) = ω1 , cof (M on) = c.
11.21 Note. The above might not strictly be true, the definition of M on in Hruˇsa´ k’s and Zindulka’s is slightly different
and it is not clear, whether they coincide.
11.22
U. Ariet Ramos-Garcia: Extremal disconnectedness and ultrafilters
(see also his talk at the 2014 Winter School)
Our work with Michael is motivated by an old question of Arhangelskii:
11.23 Question (Arhangelskii, 1967). Is there a non-discrete extremally disconnected topological group?
11.24 Theorem (Sirota, 1969). Yes, under CH.
The following is probably (the notetakers guess) from A. Louveau: Sur un article de S. Sirota, Bull. Sci. Math, 1972.
11.25 Theorem (Louveau, 1972). The group ([ω]<ω , ∆) with topology given by the neighbourhood basis of 0 specified by some
Ramsey ultrafilter U (i.e. the basis consists of {[F ]<ω : F ∈ U}) is extremally disconnected.
11.26 Definition. Given a point x ∈ X we define
!
n
n
o o
[
∗
spect(X, x) = p ∈ ω : (∃hUn : n < ωi open) p = A ⊆ ω : x ∈
Un
n∈A
S
and Spect(X) = x∈X spect(X, x). Moreover we define 0 − Spect(X, x) to consists of ultrafilters whose witnessing
sequence consists of Clopen sets and c − Spect(X) if it consits of disjoint sets.
11.27 Observation. 1. If X is extremally disconnected, then Spect(X) = 0 − Spect(X) = c − 0 − Spect(X). 2. spect(X, x)
is closed downward in the Rudin-Keisler order. 3. If X is extremally disconnected then spect(X, x) is upwards directed in the
Rudin-Keisler order.
11.28 Theorem. If X is extremally disconnected and χ(X, x) < d then spect(X, x) ⊆ {P-points}.
Proof. Assume, aiming towards a contradiction, that some p ∈ spect(X, x) is not a P-point. since X is ED we have
spect(X, x) = 0 − c − spect(X, x), i.e. there is a sequence hUn : n < ωi of disjoint clopen sets witnessing this. Since
p ∈ ω ∗ we have
[
[
Un \
Un
x∈
n<ω
n<ω
Also, since p is not a P-point there is a sequence {An : n < ω} ⊆ p with no pseudointersection in p. Let B be a local
base at x of size < d. For each B ∈ B define
n
o
[
fB (n) = max k : B ∩
Ui = ∅ ∪ {0}
i∈An ∩k
Since {An : n < ω} do not have a pseudointersection in p, we know that for each f : ω → ω
x 6∈ Hf =
[
[
Ui ,
n<ω i∈An ∩f (n)
so, in particular, there is B ∈ B which is disjoint from Hf . But then f ≤ fB . It follows that {fB : B ∈ B} form a
dominating family which contradicts the fact that |B| < d.
23
Recall the following proposition, which follows from Stone duality.
11.29 Proposition. IF X is extremally disconnected then RO(X) does not add Cohen reals iff for each continuous function
f : X → 2ω there is a nonempty open U such that f [U ] ∈ nwd(2ω ).
The previous proposition motivates the following conjecture.
11.30 Conjecture (Hruˇsa´ k). if G is an ED topological group and f : G → 2ω is continuous then there is a nonempty
open U ⊆ G such that f [U ] ∈ nwd(2ω ).
11.31 Theorem. The conjecture holds for countable Boolean ED topological groups and continuous homomorphisms.
Proof of Lemma.
11.32 Lemma. Suppose that G is a Boolean topological group. Then for any open subset U and for every neighbourhood V of 0
there is an open subset W ⊆ U such that W + W ⊆ V
Proof of Lemma. Clear.
11.33 Lemma. There is a partition U ∪ V = G \ {0} into open sets such that for each n < ω there are nonempty open
Un0 , Un1 , Vn0 , Vn1 ⊆ [sn ], where {sn : n < ω} is an enumeration of 2<ω , such that f −1 [Un0 + Un1 ] ⊆ U and f −1 [Vn0 + Vn1 ] ⊆ V
Proof of Lemma. Recursively apply the previous lemma.
Again, using lemma 1, we can construct the U, V s so that if Fn = Un1 + Un1 and En = Vn1 + Vn1 then hFn , En : n < ωi
form a cellular family on 2ω /{0}.
And then I got lost...
12
12.1
19. 2. 2014
B. Balcar & J. Verner: h,b,g (recap.)
A typical problem in mathematics is the problem of classifying structures. Since the structures in question might be
very complicated one can approach the problem by identifying some simpler relevant features of the structures and
classify them according to these features, which are sometimes called invariants. E.g. in algebraic topology, each
(pointed) space has an associated fundamental group. Spaces can then be classified according to their fundamental
groups. The fundamental group may be a very complicated object however and we can expect that it carries a lot
of information about the space. However, sometimes even very simple objects give a lot of information. E.g. the
dimension of a vector space over a given field, already carries all the information about the structure even though it
is just a number. Another example where a simple number already gives interesting information (although not all) is
the chromatic number of a graph.
This motivates the definition of cardinal invariants, which can be used to classify models of set theory according to
the properties of their real numbers. The following will only be a cursory glance at the topic, the interested reader
is advised to consult the very readable chapter on Cardinal invariants in the Handbook of Set Theory (A. Blass:
Combinatorial Cardinal Characteristics of the Continuum, chapter 6 of the Handbook of Set Theory, vol 1., Springer 2011).
Before introducing the two most well known cardinal invariants, b, d, we need some definitions.
12.2 Definition. For two sets A, B ⊆ ω we say that A ⊆∗ B if |A \ B| < ω, A = ∗B if A ⊆∗ B & B ⊆∗ A. The ∗
superscript on the relations indicates that the relations hold up to, possibly, a finite number of exceptions. Similarly,
for two functions f, g : ω → ω one defines f ≤∗ g if |{n : f (n) > g(n)}| < ω
12.3 Definition. Given a partial order (P, ≤) we say that a set X ⊆ P is unbounded if for each p ∈ P there is some
b ∈ X such that b 6≤ p. We say that it is dominating (or cofinal) if for each p ∈ P there is some d ∈ X such that p ≤ d.
Now we are ready to define the first two invariants:
24
12.4 Definition. The bounding number
b = min{B ⊆ ω ω : B is unbounded in (ω , ω), ≤∗ )
is the smallest cardinality of an ≤∗ -unbounded family of functions from ω to ω. Similarly the dominating number
d = min{B ⊆ ω ω : B is dominating in (ω , ω), ≤∗ )
12.5 Fact. ω1 ≤ b = cf b ≤ cf d ≤ d ≤ c = |R| = 2ℵ0
The last equalities are definitions and the last inequality is immediate. To see the first inequality notice that any
countable collection of functions may be dominated by a single function, which can be built inductively (this is very
similar to Cantor’s diagonal argument). Although slightly more involved, it is not very hard to prove the other
inequalities.
The following cardinal invariant was isolated in B. Balcar, J. Pelant and P. Simon: The space of ultrafilters on N covered
by nowhere dense sets, Fund. Math. 110 (1) 1980, pp. 11-24. ( where it was called simply κ)
12.6 Definition. h is the smallest cardinal κ such that the Boolean algebra P(ω)/f in is not (κ, 2)-distributive.
In forcing language, the above definition of h may be rephrased as follows: h is the smallest cardinal κ such that
forcing with P(ω)/f in adds a new subset of κ.
We will show another, equivalent, definition of h. First we need to define the notion of a MAD family. To motivate
this, note that any family of disjoint subsets of ω must necessarily be (at most) countable. What happens if we weaken
the requirement of disjointness:
12.7 Definition. A family A of infinite subsets of ω is called almost disjoint (AD for short) if each distinct A, B ∈ A are
almost disjoint, i.e. |A ∩ B| < ω. An AD family A is called maximal almost disjoint (MAD for short) if it is infinite and
cannot be extended to a larger AD family, i.e. for any infinite set X ⊆ ω there is some A ∈ A such that |X ∩ A| = ω.
The existence of MAD families easily follows from the Axiom of choice and it turns out that there always are uncountable AD families (and hence also MAD families). (For example, instead of ω consider the countable set S = 2<ω
of finite sequences of zeroes and ones. For each f ∈ 2ω let Af = {f n : n < ω} ⊆ S. Then {Af : f ∈ ω ω } is an almost
disjoint family of subsets of S of size c.)
Let us now start with some MAD family A0 . S
For each A ∈ A0 we can restrict to A and find another MAD family
AA on A. Then it is not hard to see that A1 = {AA : A ∈ A0 } will again be a mad family on ω. We can repeat this
process to eventually get a sequence {An : n < ω} of MAD families with the nice property that for each X ∈ An+1
there is some Y ∈ An such that X ⊆∗ Y (we say that An+1 refines An or, in symbols, An+1 An .
Suppose we would like to continue and define a MAD family Aω which would refine each of the previous families.
It is not immediately obvious, that this is possible. A small trick is needed. Consider some function f : ω → [ω]ω
which picks, on each level a set from An such that this set is contained in the previous choices, i.e. f (n + 1) ⊆∗
f (n) — we shall call such functions threads. Each such thread gives us a ⊆∗ descending sequence of subsets of ω.
This sequence may have an empty intersection. However (using induction) it is easily shown that there are many
infinite sets A having the property that A ⊆∗ f (n) for all n < ω (these A’s are never unique and are sometimes
called pseudointersections of the sequence. We can consider an A.D. family Af consisting of these pseudointersections.
Using the Axiom of choice, we can assume that this A.D. family is maximal (not necessarily MAD, however, no
pseudointersection can be added for it to remain A.D.). The situation can be seen in the following picture
25
Finally, we can let Aω be the collection {Af : f is a thread}. It is easy to see that Aω is an A.D. system and short
argument shows that it must be MAD. Having defined Aω we can define Aω+1 and carry on our construction, at limit
steps of countable cofinality using the above trick. What happens, however, at limit steps of uncountable cofinality
(actually at steps of cofinality ≥ t? This is not clear. Of course, there might be some suitable MAD family, which will
allow us to continue, but there might not — the above construction does not work. It turns out that, the cardinal h
can be characterized as the first cardinal where this process can stop:
12.8 Theorem. h is the least cardinal κ such that there is a refining sequence of MAD families hAα : α < κi (i.e. for α < β < κ
we have Aα Aβ ) which have no common refinement.
Since MAD families naturally correspond to tall ideals, we can also rephrase the above theorem as follows:
12.9 Theorem. h is the least size of a family of tall ideals whose intersection is not tall (or, equivalently, whose intersection is
just the ideal of finite sets).
(Recall, that an ideal I is tall, if for each infinite A ∈ [ω]ω there is an infinite I ⊆ A such that I ∈ I. The correspondance
between MAD families and ideals comes from the following observations
12.10 Observation. An AD family A is MAD iff the ideal generated by A and all finite sets is tall.
12.11 Observation. An ideal I is tall iff there is a MAD family A ⊆ I.
)
Tall ideals are ideals which are, in some sense, not too small. What if we consider a different notion of smallness.
Since ideals are subsets of P(ω) ' 2ω and since 2ω carries a natural topology given by the metric:
− min{n:f (n)6=g(n)}
2
f 6= g
ρ(f, g) =
0
f =g
we can ask whether an ideal is meager (i.e. can be covered by countably many closed co-dense sets). A simple argument shows that maximal ideals are nonmeager: If I were maximal and meager, then P(ω) = I ∪ I ∗ by maximality.
Since A 7→ ω \ A is a homeomorphism of P(ω) we have that I ∗ , the dual filter to I is also meager. However this is a
contradiction with Baire’s theorem, which – in this context — says that P(ω) is non-meager. A slightly more involved
argument shows that each ideal which is not tall must be meager. So the notion of nonmeagerness sits between tallness and maximality. It is natural to ask, whether a witnessing family consisting of nonmeager ideals for h can be
constructed. It turns out that this is not always the case, however it leads to the following definition
12.12 Definition. The idealized groupwise density number gid (= gf ) is the smallest size of a family of non-meager
ideals whose intersection is meager (or, equivalently, the ideal of finite sets).
The reason why its called idealized groupwise density is because of the related notion of a groupwise dense family
(and groupwise density number) introduced in A. Blass, C. Laflamme: Consistency Results About Filters and the Number
of Inequivalent Growth Types, J. Symb. Logic 54(1), 1989, pp. 50-56
26
12.13 Definition (Blass, Laflamme). A family G ⊆ [ω]ω is called groupwise dense if 1. it is downwards closed, i.e. if
A ⊆ G ∈ G , then A ∈ G 2. dense in ([ω]ω , ⊆∗ ), i.e. for each X ∈ [ω]ω there is an infinite
G ∈ [X]ω such that G ∈ G. 3.
S
ω
for each interval partition hIn : n < ωi of ω there is an infinite X ∈ [ω] such that n∈X In ∈ G
12.14 Definition (Blass, Laflamme). The groupwise density number g is the smallest size of a family of groupwise
dense sets whose intersection is not groupwise dense (or, equivalently, empty).
The connection between g and gid comes from the fact that, for ideals, being groupwise dense is the same as being non-meager. This follows from an old combinatorial characterization of non-meagerness for ideals discovered,
independently, by S. A. Jalaili-Naini (’76) and M. Talagrand (’80):
12.15 Theorem (Jalaili-Naini,STalagrand). An ideal I is non-meager iff for each interval partition hIn : n < ωi of ω there is
an infinite X ∈ [ω]ω such that n∈X In ∈ I.
It is immediately clear that g ≤ gid , since g considers more families. One might ask, whether there is really any
difference. It turns out that there is
12.16 Theorem (Brendle). It is consistent that g ≤ gid .
¨ Mathematik 152(3), 2007, pp. 207-215)
(see J. Brendle: Distinguishing groupwise density numbers, Monatshefte fur
It might not be clear, at first, that the cardinal invariant gid (or g) is in any way significant. However, it turns out that
it is intricately connected with the structure of filters (or ideals) on ω. To show this connection we must investigate
more closely the structure of filters. Suppose we start with ω and a filter F on
S ω. We now blow up each n ∈ ω into
a larger set Ln . The filter F on ω naturally gives rise to a filter F 0 on L =S n<ω Ln — a subset of L is in F 0 if it
contains a union of filter many blocks Ln . Formally, L ∈ F 0 iff (∃F ∈ F)( n∈F Ln ⊆ L). The process can also be
reversed.
If we have a filter F on the larger set L, we can define a ”condensed” filter F∗ on ω by declaring A ∈ F 0 iff
S
n∈A Ln ∈ F.
Suppose now we have a function f : ω → ω. This function naturally gives rise to the above situation — each number
gets ”blown-up” to its preimage, i.e. Ln = f −1 (n). The following definition says, essentially, that F ≤K G if F is a
”condensation” of G
12.17 Definition (Katˇetov, Rudin-Keisler). A filter F ≤K G if there is a function f : ω → ω such that
F = f∗ (G) = {A ⊆ ω : f −1 [A] ∈ G}.
We call f the witnessing function. If this function is finite-to-one (i.e. preimages of finite sets are finite) then we say
F ≤KB G.
12.18 Note. The above ordering, when restricted to ultrafilters, is called the Rudin-Keisler (and Rudin-Blass, respectively) ordering and denoted ≤RK (and ≤RB , respectively).
12.19 Note. It is instructive to convince onself that the definitions can be repeated for ideals and (using the JalailiNaini-Talagrand characterization) that an ideal is meager iff it is Katˇetov-Blass above the ideal of finite sets.
We have shown above that ultrafilters cannot be meager. However, if a filter is non-meager, it does not necessarily
mean that it is an ultrafilter. How far are non-meager filters from ultrafilters? The following principle says: not very
much.
12.20 Principle (Filter Dichotomy). The Filter dichotomy is the statement ”Each filter is either meager (i.e. KB above
fin) or KB above some ultrafilter”.
Is this principle true (provable in ZFC)? Is it consistent? No and Yes. Under CH, e.g., it is easy to construct a
nonmeager filter which is not KB above any ultrafilter. Showing that it is consistent is a little harder and we will not
go into the details. However, it is interesting that the FD principle can be formulated in terms of cardinal invariants.
First a definition:
12.21 Definition. The ultrafilter number u is the smallest size of an ultrafilter base (equivalently, the smallest character
of a point in ω ∗ = βω \ ω).
Now we can state the theorem, due to H. Mildenberger, which can be found in H. Mildenberger: Groupwise dense
families, Arch. Math. Logic 40, 2001, pp. 93-112:
12.22 Theorem. The FD is equivalent to u < gid
27
12.23
Honza Stary,
´ inspired by the Winter School
Let B be a complete Boolean algebra and X its Stone space. Then X is a compact extremally disconnected T2 space.
Recall that
12.24 Definition. A space X is extremally disconnected (ED for short) if the closure of each open set is clopen.
Also recall definition 11.26 of the spectrum of a point and observation 11.27 showing an equivalent definition for ED
compac spaces.
12.25 Observation. Spect(p, βω) = {q ∈ ω ∗ : q ≤RK p}
12.26 Definition. For points x, y ∈ X define x y if Spect(x, X) ⊆ Spect(y, X).
Next, recall
12.27 Theorem (Frol´ık). Each infinite ED compact space is not homogeneous.
The above theorem does not give ’honest witnesses’ (vanDouwen) to nonhomogeneity. The quest for honest witnesses motivates the following question
12.28 Question. Given the stone space S(B) of some complete Boolean algebra B, can we find x, y ∈ S(B)) with
Spect(x) 6= Spect(y)?
12.29 Definition. A point x ∈ X is discretely untouchable if it is not a limit point of each countable discrete set.
12.30 Question. What is Spect(x, X) for a discretely untouchable x?
12.31 Note. For an ED space Spect(x, X) = ∅ iff x is a P-point.
13
13.1
26. 2. 2014
P.Simon: Shelah’s proof that groupwise density is at most b+
Recall the definitions 12.14, 12.13 and 12.4.
13.2 Theorem (Shelah). g ≤ b+
Proof. see S. Shelah: Groupwise density cannot be much bigger than the unbounded number, Math. Logic Quarterly 54(4),
pp. 340-344 2008
13.3
J. Stary´
For the following recall definition 12.29.
13.4 Definition. A topological space X is homogeneous if for each x, y ∈ X there is an automorphism f : X → X such
that f (x) = y.
13.5 Theorem (Frol´ık). Let X be an infinite extremally disconnected compact space and g : X → X continuous and injective.
Then the set of fixed points F ix(g) = {x : g(x) = x} is closed and open.
13.6 Theorem (Frol´ık). If X is an infinite extremally disconnected compact space then there is an injective continuous f :
X → X such that f [X] is nowhere dense.
13.7 Corollary. Infinite extremally disconnected compact spaces are not homogeneous.
Proof. Take the f from the above theorem, let p ∈ X and let q = f (p). We shall show that if h is a homeomorphism
then h(p) 6= q. Otherwise consider g = f ◦ h and apply theorem 13.5. Then the set of fixed points of g is clopen and
this contradicts the fact that the image of f is nowhere dense.
13.8 Definition. For a complete Boolean algebra we define gc (B) to be the minimal size of a set of complete generators
of B and π(B) (also denoted πw(B) ) the minimal size of a dense subset of B. Moreover πχ(B) = min{πχ(p) : p ∈
S(B)}, where πχ(p) is the minimal size of a local π-base at p.
28
13.9 Theorem (Simon,90). Let B be a complete, ccc Boolean algebra with density π(B) ≤ c. If cf gc (B) > ω, then S(B)
contains a discretely untouchable point.
(see P. Simon: Points in extremally disconnected compact spaces, Rend. Circ. Math. di Palermo (2) 1990)
13.10 Theorem (Balcar, Simon). If πw(B) = πχ(B) then S(B) contains a discretely untouchable point.
(see B. Balcar and P. Simon: On minimal π-character of points in extremally disconnected compact spaces, Top. Appl. 41 (
1-2) pp. 133-145, 1991)
13.11 Question. Is there some B such that πχ(B) < πw(B)?
13.12 Note (E. Thuemmel). If πw(B) = πχ(B) then the same holds for Expω (B, v).
14
14.1
12. 3. 2014
J. Greb´ık: Extending asymptotic density to a measure
14.2 Definition. Given A ⊆ ω define the upper asymptotic density
d∗ (A) = lim sup
|A ∩ [1, n]|
n
d∗ (A) = lim sup
|A ∩ [1, n]|
n
n→∞
and lower asymptotic density
n→∞
14.3 Theorem (Hahn-Banach). Given a vector space V with a seminorm || · ||, W a linear subspace of V and f : W → R
a linear function such that |f (w)| ≤ ||w|| for each w ∈ W then f can be extended to a linear function g : V → R such that
f (v) ≤ ||v|| for each v ∈ V
(Recall that || · || is a seminorm if ||v|| ≥ 0,||v + w|| ≤ ||v|| + ||w|| and ||cv|| = |c|||v|| for each v, w ∈ V and each scalar c.
It is a norm if, moreover, ||v|| = 0 ≡ v = 0.)
We will apply this theorem to the above situation. We let V = {f ∈ QN : |f [N]| < ω}. Then this is a vector space and
for each A ⊆ ω its characteristic function χA is an element of V . Further, we define a seminorm on V as follows:
Pn
|f (i)|
.
||f || = lim sup i=1
n→∞
n
Let
Pn
W =
i=1
f ∈ V : lim
f (i)
n
n→∞
∈R
and define a linear functional h : W → R by letting
Pn
h(f ) = lim
n→∞
i=1
n
f (i)
.
It is clear that h(χA ) = d∗ (A) for each A ⊆ ω with χA ∈ W . Now we can use the Hahn-Banach theorem 14.3 to extend
h to h : V → R. Then for each A ⊆ ω we have h(χA ) ≥ 0 (since h(χN ) = 1 so h(χA ) + h(χAC ) = 1.
This approach to extending asymptotic density to a measure (instead of using an ultrafilter) has the advantage that
we have more control over the values of h(A) which may be chosen arbitrarily (for independent sets) between d∗ (A)
and d∗ (A).
The above is motivated by the following question:
14.4 Question. All measures are elements of [0, 1]P(ω) . For each ultrafilter U consider mU the U-extension of asymptotic density. Is the set of all finite combinations of measures of the form mU for some U dense in [0, 1]P(ω) .
14.5
J. Verner: Zindulka/Hruˇsa´ k’s question
14.6 Definition. Given a real number c ∈ R we say that a metric space (X, ρ) is c-monotonne if there is a linear order
≤ on X such that
1. d(x, y) ≤ cd(x, z) for each x ≤ y ≤ z
29
2. open intervals in the ordering are open in the metric
A metric space is monotonne if it is c-monotonne for some c ∈ R. It is σ-monotonne if it can be covered by countably
many monotonne metric spaces.
For the following, see Zindulka, O: Is Every Metric on the Cantor Set σ-Monotonne?, Real Analysis Exchange 33 (2007),
no. 2, 485–486.
14.7 Fact. If a metric space X has a dense monotonne subspace then it is monotonne.
14.8 Fact. If a metric space X is σ-monotonne then dim X ≤ 1.
In particular, [0, 1]2 is not σ-monotonne. So, under CH, there is an example of a non σ-monotonne metric space of
size ℵ1 . This motivates the following question:
14.9 Question. Is there, in ZFC, a non σ-monotonne metric space of size ℵ1 ?
14.10 Note. Hruˇsa´ k remarked that such an example cannot be separable.
14.11
M. Rubin
The following is probably known, but I would be interested to know the answer:
14.12 Question. Suppose that X is a space, d : X ×X → R+
0 a symmetric map and c > 1 such that d(x, z) ≤ c(d(x, y)+
d(y, z)) and d(x, y) = 0 ≡ x = y. Suppose, moreover, that the base of the topology of X is given by d-balls. Is X
metrizable?
14.13
J. Stary´
Recall the definition 12.24 of an extremally disconnected space.
14.14 Definition. A space is 0-dimensional if it has a base consisting of clopen sets.
14.15 Note. A T2 extremally disconnected space is 0-dimensional.
14.16 Fact. A 0-dimensional compact space is extremally disconnected iff the algebra of clopen sets is complete.
14.17 Definition. A Boolean algebra B is homogeneous if B b ' B for each b ∈ B + . A topological space X is
homogeneous if for each x, y ∈ X there is a homeomorphism f : X → X such that f (x) = y.
In this setting, recall Frol´ık’s theorem 12.27.
14.18 Note. The completion of a homogeneous Boolean algebra is always homogeneous.
14.19 Example. Borel(R)/M eager is a homogeneous complete Boolean algebra.
14.20
P. Simon, the history of Frol´ık’s theorem
14.21 Theorem (Frol´ık). If X is extremally disconnected and compact and f : X → X is an embedding. Then we can partition
X into four clopen sets X = C0 ∪ C1 ∪ C2 ∪ C3 such that f C0 = id and F [Ci ] ⊆ Cj ∪ Ck for 1 ≤ i 6= j 6= k 6= i ≤ 3.
The consequence of the above theorem is that the fixed points of embeddings of an extremally disconnected space
form a clopen set in the space.
14.22 Example. Consider βω1 . This is an extremally disconnected space. Consider the equivalence relation U1 ∼ U2 if
both are uniform (i.e. do not contain countable sets). Then a bijection f : ω1 → ω1 of limit and successor ordinals generates a homeomorphism of βω1 whose fixed-points are only uniform ultrafilters. So βω1 / ∼ has an automorphism
which has only a single fixed point, in particular, by the above theorem, cannot be extremally disconnected.
Frol´ık’s theorem 12.27 is a consequence of the following (due to Balcar and Franˇek ?) and 14.21.
14.23 Theorem (Frol´ık, 1967). If X is an infinite extremally disconnected space contains a nowhere dense copy of itself.
30
Proof of 12.27. Let X be infinite extremally disconnected and Y ⊆ X a nowhere dense copy of X with h : X ' Y .
Choose x ∈ X \ Y such that h(x) ∈ Y . If X were homogeneous, there would be a homeomorphism f : X → X such
that f (y) = x. Then y is a fixed point of the mapping h ◦ f . However, h ◦ f [X] ⊆ Y which is nowhere dense, so the
set of its fixed points cannot be clopen — a contradiction with 14.21.
14.24 Note. Frol´ık’s proof, apparently (see W. W. Comfort: Some recent applications of ultrafilters to topology, General
Topology and Its Relations to Modern Analysis and Algebra IV, Lecture notes in mathematics vol. 609, Springer 1977,
pp 34-42), was different and used a result of K. Kunen that there are incomparable ultrafilters on ω.
All of the above was motivated by the following considerations. The algebra P(ω)/f in is homogeneous. How about
its Stone space, ω ∗ ?
14.25 Theorem (W. Rudin, 1956). Under CH, ω ∗ contains a P-point so, in particular, is not homogeneous.
14.26
J. Stary:
´ Ultrapowers of Boolean algebras
Let B be a complete Boolean algebra (cBA for short) and A an algebra. Consider the set of ”B-names”
F = {f : P →
W
A : P ⊆ B is a partition of 1} for elements of A. Given an ultrafilter U on B we say f =U g if {r ∈ dom(f ) ∩ dom(g) :
f (r) = g(r)} ∈ U. =U is an equivalence relation and the set F naturally carries a structure of a Boolean algebra, it
will be denoted AB /U and called the Boolean ultrapower of A.
14.27 Note. Consider, moreover, the set P art(B) of partitions of 1 in B. Then the partial ordering of refinement
makes P art(B) a directed set. Moreover AB /U is a direct limit of {AP /UP : P ∈ P art(B)} where UP is the ultrafilter
on P generated by U
Now let B be a ccc, cBA. For an infinite P ∈ P art(B) denote BP to be the algebra generated by P . Then, if P is
infinite, BP is isomorphic to P(ω).
14.28 Fact. B is a direct limit of the system {BP : P ∈ P art(B)} (again using the refinement ordering on partitions).
Aiming to define an ultrafilter sum, for b ∈ B we say b ∈
ω}.
P
UP
Fn if {n ∈ ω : b ∩ pn ∈ Fn } ∈ UP , where P = {pn : n <
14.29 Definition. An ultrafilter U on a ccc atomless BA B is discretely touchable if
Pthere is a partition P ∈ P art(B),
P = {pn : n < ω}, with P ∩ U 6= ∅ and ultrafilters Vn with pn ∈ Vn such that U = UP Vn .
14.30 Conjecture. If B is a ccc, atomless cBA of weight ≤ c then St(B) it contains a discretely touchable ultrafilter.
15
15.1
19. 3. 2014
D. Chodounsky:
´ Mathias forcing (Part I.)
The following is joint work with L. Zdomskyy, D. Repovˇs, based on D. Chodounsky,
´ D. Repovˇs, L. Zdomskyy:
Mathias Forcing and Combinatorial Covering Properties of Filters, preprint 2014
15.2 Definition. Given a filter F on ω we define Mathias forcing with respect to F:
MF = {(s, F ) : s ∈ [ω]<ω , F ∈ F},
where (s, F ) ≤ (t, H) if t v s, F ⊆ H and s \ t ∈ H.
15.3 Note. Mathias forcing is ccc (no uncountable antichains), even σ-centered since conditions having the same first
coordinate are compatible.
We will be interested in the following properties with respect to forcing extensions by Mathias forcing.
15.4 Definition. Given a model M of ZFC we say its extension N ⊇ M is ω ω-bounding if each function f ∈ N ∩ ω ω
is coordinate-wise bounded by some function d ∈ M ∩ ω ω. It is almost- ω ω-bounding if each unbounded subfamily of
(ω, ≤∗ ) in M remains unbounded in N . We say that N contains a dominating real if there is a d ∈ N ∩ ω ω such that for
each f ∈ ω ω ∩ M we have f ≤∗ d, i.e. |{n : f (n) > d(n)}| < ω.
15.5 Note. The above definition of almost ω ω-bounding is not standard; It is proved in the cited preprint that it is, in
fact, equivalent to the standard definition, given below.
31
15.6 Definition. A forcing P is almost ω ω-bounding if for each condition p ∈ P and each name f˙ for a function from ω
to ω there is a function g : ω → ω such that for each A ∈ [ω]ω there is a stronger q ≤ p such that q f˙ A 6≥ g A.
One part of the following equivalence is Lemma 3 in the referenced paper.
15.7 Proposition. A forcing P is almost ω ω-bounding iff forcing extensions by P are almost ω ω-bounding.
Proof. Assume P is almost ω ω-bounding, U ⊆ ω ω an unbounded family and, aiming towards a contradiction, f˙ a
is a P -name for a function from ω to ω and p ∈ P a condition forcing that dotf dominates U . Since P is almost
ω
ω-bounding there is a g : ω → ω satisfying the conditions from the definition. Since U it is unbounded, there is
A ∈ [ω]ω and h ∈ U such that g A ≤ h A. By assumption there is a condition q ≤ p and a name B˙ for an infinite
subset of A such that q f˙ B˙ ≤ g B˙ ≤ h B˙ contradicting the fact that p forces that f˙ dominates U .
Assume, on the other hand, that P is not almost ω ω-bounding. So there is a p ∈ P and a P -name f˙ for a function
from ω to ω such that for each g : ω → ω there is an infinite Ag ∈ [ω]ω such that p g Ag ≤ f˙ Ag . Then the family
U = {g Ag ∪ 0 (ω \ Ag ) : g ∈ ω → ω} is an unbounded family of functions in the ground model which is bounded
by f˙ in the extension.
15.8 Observation. Mathias forcing cannot be bounding.
15.9 Observation. The generic real for MF is a pseudointersection of F.
In particular if the filter F is rapid (i.e. the enumeration functions of elements of F form a dominating family) the
generic real is a dominating real. If F is nonmeager (which, in particular, implies that the family of enumeration
functions of elements of F is unbounded) then MF is not almost ω ω-bounding.
The motivation for the following topological definitions which characterize MF being almost ω ω-bounding and not
adding dominating reals is as follows. A necessary condition for both properties is that F itself (or its enumeration
functions) is bounded. However it might happen that F is bounded however a continuous image by f of F ⊆ 2ω
might be unbounded. Then it is reasonable to expect that the f image of the generic real will contradict ω ω bounding.
15.10 Definition. A topological space X is Hurewicz if every continuous image of X in ω ω is bounded (i.e. a bounded
family of functions in (ω ω, ≤∗ )). It is Menger if no continuous image of X in ω ω is dominating.
15.11 Note. σ-compact spaces are Hurewicz.
15.12 Note (probably). Subsets of 2ω are Hurewicz iff they are strong measure zero.
15.13 Theorem (Arhangelskii, ?). Analytic subsets of 2ω are Hurewicz iff they are Fσ , i.e. σ-compact.
15.14 Note. Preservation of these properties under products (or powers) is an area of active research.
The following theorem gives the standard definition of Menger property.
15.15 Theorem. A space X is Menger iff for
S each sequence hUn : n < ωi of open covers of X there is a sequence hFn : n < ωi
such that Fn ∈ [Un ]<ω for each n < ω and n<ω Fn is a cover of X.
To rephrase the Hurewicz property in a similar language we need some additional notions concerning covers.
15.16 Definition. An open cover U of a space X is an ω-cover if for each F ∈ [X]<ω there is a U ∈ U such that F ⊆ U ,
i.e. not only points are covered but even finite sets.
15.17 Observation. If U is an open cover then all finite unions of elements of U form an ω-cover.
15.18 Definition. An open cover U is a γ-cover if for each x ∈ X the set {U ∈ U : x 6∈ U } is finite.
A typical example of a γ-cover comes from writing the space as an increasing union of open sets.
We are now ready to rephrase the Hurewicz property.
15.19 Theorem. A space X is Hurewicz iff S
for each sequence hUn : n < ωi of ω-covers of X there is a sequence hFn : n < ωi
such that Fn ∈ [Un ]<ω for each n < ω and { Fn : n < ω} is a γ-cover of X.
Proof. We shall only show that a X ⊆ ω ω satisfying the alternative condition is bounded. Let Uk = {Unk : n < ω}
where Unk = {f ∈ X : f (k) < n}. Apply the condition to find a γ-cover. This γ-cover gives a bound on X, i.e. via
d(k) = max{n : Unk ∈ Fk }.
15.20 Notation. Given X ⊆ ω let ↑ X = {Y ⊆ ω : X ⊆ Y }.
32
15.21 Note. ↑ X is a closed set which is open iff X is finite.
The following is core lemma for dealing with Hurewicz and Menger properties of filters.
15.22 Lemma. Given a filter F and an open cover O of F ⊆ 2ω there is a Q ⊆ [ω]<ω such that
[
[
F⊆
↑q:q∈Q ⊆
O.
Proof. We are given O and we may assume that O consists only of basic open sets. Fix an F ∈ F. Then ↑ F is a
closed set, hence it is compact so there is a finite O0 ⊆ O covering ↑ F . We may assume that O0 = {[sn ] : n < k} with
|sn | = |sm | = N for each n, m < k. Now pick n < k such that F ∩ N = sn = sF . Then ↑ F ⊆↑ sF .
Finally let Q = {sF : F ∈ F}.
15.23 Corollary. When checking the menger property for filters we need only check open covers consisting of elements of the
form ↑ q for a finite q ∈ [ω]<ω .
Due to lack of time, we will only recall the following definitions which Hruˇsa´ k et al. use to deal with Mathias forcing.
15.24 Definition. Given a filter F on ω we define F <ω to be a filter on [ω]<ω \ {∅} generated by sets of the form [F ]<ω
for F ∈ F.
15.25 Observation. A set X ⊆ [ω]<ω is positive with respect to (F <ω )+ iff {↑ x : x ∈ X} is an open cover of F.
15.26 Definition. A filter F on a countable set S is P + iff for each ⊆-descending sequence hXn : n < ωi of F-positive
sets there is an F-positive X which is almost contained in each Xn .
15.27 Definition. Given a filter F on ω we let F <ω be the filter on [ω]<ω \ {∅} generated by sets of the form [F ]<ω ,
where F ∈ F.
15.28 Theorem (Hruˇsa´ k,Minami). MF does not add dominating reals iff F <ω is a P+ -filter.
Given that the above definition may be reformulated as follows
15.29 Observation. A S
filter F is P+ iff for any sequence hXn : n < ωi of F-positive sets ther is a sequence hYn : n < ωi with
<ω
Yn ∈ [Xn ] such that n<ω Yn is F-positive.
this, together with Hruˇsa´ k’s and Minami’s result, almost immediately gives a topological reformulation of the property that MF does not add dominating reals.
16
26. 3. 2014
´ Yasunao Hattori (Shimane University,
16.1 Announcement. Tomorrow at 10:40 in Karl´ın, seminar room of the MU,
Matsue, Japan) will talk about The Separation Dimension and Infinite-Dimensional Spaces.
16.2
W. Kubis: MAD families on singulars
The following is based on Kojman, M., Kubis, W. and Shelah, S.: On two problems of Erd¨os and Hechler: New Methods in
Singular Madness, Proc. Amer. Math. Soc. 132 (2004), no 11, 3357-3365
16.3 Definition. Let S be a set of size µ. A family A ⊆ [S]µ is called µ-almost disjoint if for |A ∩ B| < µ for distinct
A, B ∈ A. It is maximal µ-almost disjoint (µ-MAD for short) if it is µ-almost disjoint and it is maximal w.r.t. inclusion
with this property.
16.4 Note. A partition of S into < cf µ many sets is always a maximal µ-almost disjoint family.
For the remainder of this talk fix ℵ0 ≤ κ = cf µ < µ and say that a µ-MAD family A is nontrivial if |A| ≥ κ
16.5 Excercise. Use standard diagonalization to show that there is no µ-MAD family of size cf µ.
16.6 Definition. The (MAD) spectrum of µ is defined to be the set M AD(µ) = {|A| : A is nontrivial µ − M AD}
16.7 Note. It is clear that M AD(µ) ⊆ [κ+ , 2µ ]
33
16.8 Proposition. M AD(cf µ) ⊆ M AD(µ). Proof: Let A be a κ = cf µ-mad family on κ. Fix an increasing sequence
hµα : αS< κi of regular cardinals with supremum µ and partition µ into sets Mα such that |Mα | = µα . Finally for A ∈ A let
BA = α∈A Mα . Then {BA : A ∈ A} is a µ-mad family on µ.
16.9 Note. The above also follows from the fact that the Boolean algebra P(κ)/[κ]<κ regularly embeds into P(µ)/[µ]<µ .
16.10 Corollary. 2κ < µ ⇒ aµ ≤ 2κ .
16.11 Corollary. 2<ℵω < ℵω ⇒ 2ℵω ∈ M AD(ℵω ) Proof: Take the full binary tree of height ℵω and extend it to a ℵω -MAD
family (on the set of nodes).
16.12 Question (Comfort). When does µ ∈ M AD(µ)?
¨ Hechler). Is it consistent that µ 6∈ M AD(µ)?
16.13 Question (Erdos,
¨ Hechler). Does 2κ < µ imply µ ∈ M AD(µ)?
16.14 Question (Erdos,
16.15 Theorem. [κ+ , µ] ∩ M AD(µ) is either empty or is of the form [aµ , µ] for aµ = min M AD(µ).
Proof of theorem.
16.16 Lemma. Assume hλα : α < θi ⊆ M AD(µ) with cf θ = θ < λ0 and λ = sup{λα : α < θ} singular. Then
λ ∈ M AD(µ).
Proof of lemma. Let A0 be a µ-mad family of size
S λ0 . Take the first θ-many sets {Aα : α < θ} of A0 and on each Aα
pick a µ-mad family Bα of size λα . Then B = α<θ Bα is a µ-mad family of size λ.
16.17 Lemma. Every regular λ ∈ [aµ , µ) is in M AD(µ).
Proof of lemma. First notice that λ > κ. Let aµ < λ = cf λ < µ. For each δ < λ of cofinality κ (i.e. δ ∈ Sκλ ) fix a
δ
continuous increasing hlα
: α < κi sequence with supremum δ and l0δ = 0. Notice, that whenever we partition µ into
κ many pieces of size κ we can find a µ-mad family of size aµ which contains the pieces. Using this we fix Aδ a µ-ad
δ δ
family on µ × λ of size aµ such that
S Aδ ∪ {µ × [lα , lα+1 ) : α < κ} is µ-mad on µ × [0, δ). Then Aδ0 ∪ Aδ1 is µ-ad for
λ
each δ0 < δ1 ∈ Sκ . Now let A = δ∈Sκλ Aδ . This is µ-a.d.
16.18 Claim. A ∪ {µ × {α} : α < λ} is µ-mad.
Proof of claim. Fix X ⊆ µ × λ of size µ. Then there is a minimal δ < λ such that |X ∩ µ × δ| = µ. It is not hard to see
that δ ∈ Sκλ .
Recall the definition of the bounding number for regular cardinals:
16.19 Definition. bκ = min{|B| : B ⊆ κκ is unbounded in <∗ }, where f <∗ g ≡ |{α : f (α) ≥ g(α)}| < κ.
For singular cardinals it is convenient to modify the definition slightly:
16.20 Definition.
bµ = sup{b(
Y
µi , <∗ ) : µi = cf µi & µi % µ}
i<κ
16.21 Theorem. min{bκ , bµ } ≤ aµ
Proof. Assume A is µ-a.d. on κ × µ ofQ
size < min{bκ , bµ }. Fix an increasing sequence hµα : α < κi of regular cardinals
with supremum µ such that |A| < b( α<κ µα , <∗ ). Let B = A \ {{α} × µ : α < κ}. For each B ∈ B define fB : κ → κ
by letting fB (γ) = min{α : |B ∩ {γ} × µ| < µα }. Now there is a function f : κ → κ dominating the fB s such that
|B∩({α}×µ for each B
Q∈ B. Then given B ∈ B the intersection B∩({γ}×f (γ)) is bounded for all but boundedly many
γs,Qi.e. there is gB ∈ α<κ
Q µf (α) such that for all but
Q boundedly many γs we have B ∩ ({γ}S× f (γ)) ⊆ gB (γ). Since
b( α<κ µf (α) , <∗ ) ≤ b( α<κ µα , <∗ ) there is g ∈ α<κ µf (α) dominating the gB s. Let X = α<κ {α} × [g(α), f (α)).
Then X is almost disjoint from each A ∈ A so A is not µ-mad.
16.22 Corollary. Martin’s axiom with 2ℵ0 > ℵω implies that there is no ℵω -MAD family of size ℵω .
34
16.23 Theorem. If aµ < bµ then [aµ , bµ ) ⊆ M AD(µ). Moreover, if bµ is a successor of a singular then bµ ∈ M AD(µ).
16.24 Question. Can aµ be singular?
16.25 Note. Brendle proved that a can be singular, even of countable cofinality. (see Brendle, J.: The Almost-Disjointness
Number May Have Countable Cofinality, Trans. Amer. Math. Soc. 355(7), 2003)
Q
16.26 Definition. A λ-scale in ( α<µ µα , <∗ ), where hµα : α < κi, is Q
an increasing sequence of regular cardinals with
κ < µ0 , is a <∗ -increasing sequence hfγ : γ < λi which is cofinal in ( α<µ µα , <∗ ).
Q
16.27 Theorem (Shelah). There is a set M ∈ [ω]ω such that ( n∈M ℵn , <∗ ) has an ℵω+1 -scale.
Q
16.28 Theorem (Shelah). For each λ ∈ [µ+ , bµ ) there is a (continuous) λ-scale in ( α<µ µα , <∗ ) for a suitable sequence µα .
16.29 Note. If bµ is a successor of a regular then there is also a bµ -scale. Also, there is always a µ+ -scale.
The proof of theorem 16.23 heavily uses Shelah’s PCF theory, in particular theorem 16.28. Then it roughly follows
δ δ
the proof of where the role of the ”slices” µ × [lα
, lα+1 ) is taken by the space between the functions in the scale, i.e.
S
[f
(γ),
f
(γ)).
α+1
γ<κ α
This has a connection to:
16.30 Theorem (Balcar, Simon). P(κ)/[κ]<κ collapses bκ to ω for regular κ > ω.
16.31 Theorem (Balcar, Simon). If µ is singular of countable cofinality, then P(µ)/[µ]<µ is σ-closed and collapses µℵ0 to ω1 .
Shelah proved:
16.32 Theorem (Shelah). If λ ∈ M AD(κ) \ κ+ then P(κ)/[κ]<κ collapses λ to ω (again, κ > ω is regular).
17
17.1
2. 4. 2014
M. Fabi´an: Separable reductions and rich families in variational analysis
17.2 Setting. Let X be a metric space (or a normed linear space). We will consider the family S of all closed separable
subspaces of X.
17.3 Definition. Given a function f : X → R, let Cf denote the family of Y ∈ S such that if f Y is continuous at
some point x of Y then f is continuous at x.
17.4 Proposition. Cf is cofinal in (S, ⊆).
Proof. Fix Z ∈ S. We shall construct a sequence of countable sets hCn : n < ωi. Fix C0 arbitrarily such that C0 = Z.
Assume we have constructed Cn . For each c ∈ Cn and q ∈ Q+ findSa countable Dc,q ⊆ B(c, q) with c ∈ Dc,q ,
diam(Dc,q ) ≤ q and diam(f [Dc,q ]) = diam(f [B(c, q)]). Next let Cn+1 = c∈Cn ,q∈Q+ Dc,q (in case of linear spaces, we
finish by taking the ”rational span” of the union, i.e. all finite linear combinations .with rational coefficients). Finally
S
let Y = Cn . It is not hard to see that this Y works.
The following notion was defined in J. Borwein, W. Moors: Separable Determination of Integrability and Minimality of
the Clarke Subdifferential Mapping, Proc. AMS, 128(1) 1999, pages 215-221.
17.5 Definition (Borwein, Moors). A family R ⊆ S is rich if
1. it is cofinal
2. closed under closures of countable increasing unions ( σ-complete)
17.6 Note. This corresponds to the notion of a ( σ-)club in a partially ordered directed set.
17.7 Observation. The intersection of countably many rich families is rich.
17.8 Definition. Let
R = {Y ∈ S : (∀q ∈ Q+ , y ∈ Y )(diamf [B(y, q)] = diamf [B(y, q) ∩ Y ])}
17.9 Observation. R is rich in S.
35
The above observation can also be vaguely formulated saying: ”continuity can be separably reduced”.
17.10 Theorem (Lindenstrauss, 1965). Let X be a reflexive Banach space (i.e. X ∗∗ ' X ). Then there is a PRI sequence
hPα : α < d(X)i of retractions (where d(X) is the density of X) satisfying some conditions.
(see J. Lindenstrauss: On reflexive spaces having the metric approximation property, Israel J. Math, 3(4), 1965, pages 199204)
The following is a generalization due to W. Kubis in Banach spaces with projectional skeletons, J. Math. Anal. Appl.
350(2), 2009, pages 758-776.
17.11 Definition. Given two linear functionals p, q ∈ L(X) on a Banach space X we define p q if p[X] ⊆ q[X].
17.12 Theorem (Kubi´s). There is P ⊆ L(X) system of projections such that
1. p ◦ p0 = p for each p p0 ∈ P,
2. p[X] is separable for each p ∈ P
3. P is rich in (L(X), ).
17.13
J. Greb´ık: Ultrafilter Question
∗
Is
S there an easy way to show that there is an ultrafilter U ∈ ω such that for each U ∈ U there is a k ≥ 3 with
n∈U [n(k − 1), nk] ∈ U. The only proof I know uses idempotent ultrafilters.
17.14
D. Chodounsky:
´ Filter Mathias forcing (Part II.)
For Part I. see section 15.1. Recall definitions 15.10 (Menger and Hurewicz properties), 15.18 (γ-cover) and 15.16
(ω-cover). Also recall the corollary 15.23 useful for checking the Hurewicz property for filters.
Also recall definition 15.27 and theorem 15.28.
17.15 Theorem (Chodounsky,
´ Repovˇs, Zdomskyy). Assume F is Hurewicz. Then MF is almost ω ω-bounding.
Proof. Assume not. So let X ⊆ ω ω be an unbounded family of functions and g˙ : ω → ω is a name for a function
dominating X. Hence for each f ∈ X there is some condition (sf , F f ) ∈ MF and nf < ω such that
(sf , F f ) (∀n ≥ nf )(f (n) ≤ g(n)).
˙
If we partition X into countably many pieces one of the pieces will be unbounded. In particular, we may WLOG
assume that there is s∗ , n∗ such that sf = s∗ and nf = n∗ for each f ∈ X. For m ∈ ω define
Sm = {s ∈ [ω]<ω : (∃Fs ∈ F, gs ∈ ω)((s∗ ∪ s, Fs ) g(m)
˙
= gs )}
It is easy to see that Sm ∈ F <ω ., i.e. {↑ s : s ∈ Sm } is a cover
S of F (i.e. an ω-cover of F, sinceSF is a filter). We now use
that F is Hurewicz to choose Um ∈ [Sm ]<ω such that { Um : m < ω} is a cover of F, i.e. m<ω Um is F <ω -positive.
Define d(m) = max{gs : s ∈ Um }. We will show that d dominates X which will be the required contradiction. Pick
some f ∈ X. Then there is m0 < ω such that for each m ≥ m0 there is sm ∈ Um such that sm ⊆ F f . Now pick
n > n∗ , m0 . Then
(s∗ ∪ sn , Fsn ∩ F f ) f (n) ≤ g(n)
˙
≤ d(m),
i.e. f (n) ≤ d(n) for all n > m0 , n∗ , i.e. d dominates f .
A similar proof gives:
17.16 Theorem (Chodounsky,
´ Repovˇs, Zdomskyy). Assume F is Menger. Then MF does not add dominating reals.
17.17 Note. The above theorems can be reversed, the suitable unbounded (or dominating) family which will be
destroyed by MF can be defined from a sequence of covers of F violating the Hurewicz (or Menger) property. The
dominating/unbounded function in the extension can be easily defined from the generic real (since each cover from
the witnessing sequence covers the generic real).
36
18
9.4.2014
18.1 Announcement. Next week, Tue 15. 4. , there will be a workshop in Banach spaces organized by Wieslaw Kubi´s
and Jerzy Kakol at the Mathematical institute.
18.2 Announcement. Wed, 23. Apr, Honza Stary´ will have his thesis defense at 11 am. Afterwards, at 12:30, we will
go to the Green Garden restaurant.
18.3
J. Greb´ık: Ultrafilters and dynamical systems
18.4 Definition. A dynamical system with discrete time is a space X together with a continuous map f : X → X (or,
possibly, more maps).
18.5 Note. The intention is that the map transforms the space X in units of time, i.e. a point x ∈ X is moved to
f n (x) = f (f (· · · f (x) · · · )) in time n < ω.
| {z }
n×
18.6 Definition. A point x ∈ X is a fixed point if f (x) = x, it is periodic if there is n < ω such that f n (x) = x, it is
almost periodic (or recurrent) if for each open neighbourhood U of x there is n < ω such that f n (x) ∈ U . It is strongly
almost periodic if for each open U there is dU < ω such that for each n < ω the set {f n (x), f n+1 (x), . . . , f n+dU (x)} ∩ U
is nonempty.
18.7 Theorem. If X is compact then it contains a strongly almost periodic point.
18.8 Definition. Two points x, y ∈ X, where X is a metric space, are proximal if for each ε > 0 there is an n < ω such
that %(f n (x), f n (y)) < ε
We will now consider a dynamical system where the space is composed of ultrafilters on natural numbers and the
maps are derived from addition and multiplication on natural numbers.
18.9 Definition. Given A ⊆ N let A − 1 = {n ∈ N : n + 1 ∈ A}. Given U an ultrafilter on N let sh(U) = {A : A − 1 ∈ U}.
18.10 Definition. Given a sequence hpn : n < ωi ⊆ βN and an ultrafilter U there is p ∈ βN such that p = U − lim pn
18.11 Observation. Given U, V ∈ βN it is not hard to see that
n
o
V − lim shn (U) = A ⊆ N : {n < ω : A − n ∈ U} ∈ V
18.12 Definition. For U, V we define
U + V = V − lim shn (U),
and
n
o
U · V = V − lim n · U = A : {n : A/n ∈ U} ∈ V ,
18.13 Note. The + operation on βN, the set of ultrafilters on N, is
1. associative
2. its restriction to ω is the standard addition
3. given n < ω then U + n = n + U.
4. is continuous in the right coordinate, i.e. the map U 7→ U + x is continuous for each U ∈ N.
18.14 Lemma (Ellis-Numakura). Each (X, ·) compact, T1 , right-semi-topological semigroup contains an idempotent, i.e. an
element x ∈ X such that x · x = x.
Proof. Let K ⊆ X be the system of compact sub-semigroups such that K · K = K for each K ∈ K ordered by
inclusion. Then K is nonempty since X ∈ K. Let K be a minimal element of K. Take e ∈ K. Then e · K is a compact
subsemigroup of K so, by minimality of K, e·K = K and there is s ∈ K such that e·s = e. Consider S = {s : e·s = e}.
This is a compact subsemigroup of K so, again by minimality, S = K. It follows that e · e = e.
18.15 Theorem (Kal´asˇ ek). There is no additive idempotent which would be, at the same time, a multiplicative idempotent.
(see his thesis)
18.16 Definition. (X, f ) is a subsystem of (Y, f ) if X is a closed invariant subset of Y .
37
18.17 Observation. Every compact dynamical system contains a minimal dynamical subsystem.
18.18 Observation. Every two different minimal subsystems are disjoint.
S
18.19 Theorem (Hindman).
Suppose [ω]<ω = i<k Ai . Then there is j < k and D ∈ [Aj ]ω consisting of disjoint sets such
S
that ∀F ∈ [D]<ω F ∈ Aj .
<ω
18.20 Question (Chodounsky).
´ Is there an interval partition hIn : n < ωi such that
S for each partition
S [ω] = A0 ∪ A1 ,
ω
<ω
there is i < 2 and D ∈ [Aj ] consisting of disjoint sets such that ∀F ∈ [D]
F ∈ Aj and D intersects all but
finitely many interval In .
18.21 Note.
1. If the answer is no then each union ultrafilter has a non-meager core and is almost ordered.
2. Even a consistent answer (either way) would be interesting.
3. Todorˇcevi´c heard the problem and his opinion is that the answer is no.
19
19.1
16.4.2014
J. Greb´ık: Extending density to all sets of integers.
Recall that, for A ⊆ N, we define the density of A
d(A) = lim
n→∞
|A ∩ [1, n]|
n
if it exists. We let D be the family of all sets which have a defined density. We let M denote the family of all measures
on P(N) extending d.
Let V = {f | f : N → R, |rng(f )| < ω} and let
Pn
o
n
i=1 f (i)
.
U = f : ∃ lim
n→∞
n
If we apply the Hahn-Banach theorem to a function from U , we will necessarily get a shift-invariant measure. Moreover, this measure will be very regular, in particular, if m is such a measure, A ⊆ N and ϕ : A → N is one-to-one, then
for each B ∈ D we have m(ϕ−1 [B]) = m(A) · d(B), i.e. m ”respects” relative densities.
If we take a different approach we can define, for U ∈ ω ∗ ,
dU (A) = U − lim
|A ∩ [1, n]|
.
n
Then each dU will be an element of M1 . We let MU = {mU : U ∈ ω ∗ } ⊆ M1 .
We let M2 be the closure (in [0, 1]P(N) ) of the convex hull of MU .
19.2 Fact. M1 and M2 are both convex and closed (in [0, 1]P(N) ).
19.3 Observation. M2 ⊆ M1 ⊆ M.
19.4 Question. Are the above inclusions strict?
Let Mσ = {m ∈ M : m is σ − additive}.
19.5 Question. Is Mσ a closed subset of [0, 1]P(N) ?
19.6 Definition. Given han : n < ωi, the Caesar-limit of this sequence is, if it exists,
Pn
i=0 an
.
lim
n→∞
n
19.7 Question. When is dU σ-additive?
19.8 Definition. An ultrafilter is ∗-invariant if for each U ∈ U there is k ≥ 2 such that
[
[k · n, (k + 1) · n] ∈ U
n∈U
19.9 Theorem. The measure dU is σ-additive iff U is not ∗-invariant.
38
19.10 Definition. A set A ⊆ ω is thin if
lim
n→∞
eA (n)
= 0,
eA (n + 1)
where eA is the (increasing) enumeration of A.
19.11 Theorem. If an ultrafilter U contains a thin set, then it is not ∗-invariant so, in particular, dU is σ-additive.
19.12 Question. Let T be the set of ultrafilters contining a thin set. Consider the set of ∗-invariant ultrafilters as a
subspace of ω ∗ \ T . What are its properties. Note that ω ∗ \ T is a closed nowhere dense subset of ω ∗ .
19.13 Definition. An U ultrafilter is thin if there is a U ∈ U such that
lim sup
n→∞
eU (n)
< 1,
eU (n + 1)
where eU is the (increasing) enumeration of U .
19.14 Proposition. There is a function G : ω ∗ → {U : U is thin} such that for each U we have dU = dG(U ) and, moreover, U
is ∗-invariant iff G(U) is ∗-invariant.
Proof of theorem 19.9. Assume U is not ∗-invariant and let U be a witness. For each k ≥ 2 we know that
[
[k · n, (k + 1) · n] 6∈ U,
n∈U
so also
[
[k · 2n, (k + 1) · 2n] 6∈ U,
n∈U
By the previous proposition we may assume that U is thin. Then there is some n < ω such that
eU (n)
< 1/2.
eU (n + 1)
Define
Λ : P(N) →
∞
Y
P([1, i])
i=1
by letting Λ(A)(n) = A ∩ [1, n]. Similarly let
ΛU : P(N)/dU →
∞
Y
P([1, i])/mU ,
i=1
where
mU (f ) = U − lim mn (f (n)),
and mn is the normalized counting measure on P([1, n]).
19.15 Sidenote. Consider P(N)/dU as a metric space (with %(A, B) = dU (A4B)) and the target of ΛU as a metric
space. Then ΛU gives rise to an isometry between these two spaces.
We shall show that ΛU is onto. This will suffice to show that dU is σ-additive since the target space of ΛU together
with its measure is σ-additive and ΛU is measure preserving.
Q∞
So let f ∈ i=1 P([1, i])/dU . We will find X ⊆ N such that mU (ΛU (X)4f ) = 0. Let
[
X=
[eU (n), eU (n + 1)] ∩ f (eU (n).
n<ω
We will show that this X works. To do this, we will find a sequence hUk : k < ωi ⊆ U satisfying
mU (f (n)4Λ(X)(n)) < 1/k
for each k < ω, n ∈ Uk . Let
Uk = U \
[
[k · 2n, (k + 1) · 2n]
n∈U
.
The other implication follows along the lines of Fremlin’s proof.
39
20
30. 4. 2014
20.1
J. Stary:
´ Pi-characters
20.2 Definition. Let X be a topological space. We say that a family of nonempty open sets B is a local π-base at x ∈ X
if for each open U containing x there is B ∈ B such that B ⊆ U . We denote πχ (x) the minimal cardinality of a local
π-base at x and call it the π-character of x. We let πχ (X) = min{πχ (x) : x ∈ X}. For a Boolean algebra B we let
πχ (B) = πχ (st(B)). When we talk about the π-weight of Boolean algebra πw(B), we mean the π-weight of its stone
space.
The following is proved in B. Balcar, P. Simon: On minimal pi-character of points in extremally disconnected compact spaces,
Top. Appl. 41(1,2) 1991, pp. 133-145.
20.3 Theorem (Balcar,Simon). Let B be a complete, ccc Boolean algebra which is homogeneous in πw. If πχ (B) = πw(B) then
St(B) contains a discretely untouchable point.
20.4 Question. Is there a ccc, complete Boolean algebra for which πχ (B) < π(B)?
20.5 Note. Mary Bell proved that the answer is yes if we drop the completeness requirement.
20.6 Theorem. If B is a complete, ccc, atomless Boolean algebra such that d(St(B)) ≤ πχ (B). Then either it contains two
points with different π-characters or the minimal π-character is actually equal to πw(B)
20.7 Definition. Let X be a topological space. Its tightness, t(X), is the smallest κ such that whenever some x ∈ X is
in the closure of a set A ⊆ X then it is already in the closure of a subset B ⊆ A of size κ.
20.8 Observation. If t(X) ≤ d(X) then any dense set contains a dense subset of size d(X).
20.9 Example. [0, 1]ω1 is a separable compact space whose tightness is uncountable, since the space is universal for
Tychonoff spaces of weight at most ω1 , in particular it contains a copy of ω1 + 1.
20.10 Question. Is there an EDC space X with t(X) > d(X).
20.11 Note. Yes, βω, since it contains weak P -points guaranteeing uncountable tightness. If we want a space without
ˇ
´
isolated points, start with the space from Dow, Gubbi, Szymanski
taking a weak P-point as the parameter. The CechStone compactification of this space is a separable, EDC space without isolated points and uncountable tightness
(since the space has an ℵ0 -bounded remainder).
´
(For the space see A. Dow, Gubbi, A. V. Gubbi, A. Szymanski:
Rigid Stone spaces within ZFC, Proc. Am. Math. Soc.
102(3), 1988.)
20.12 Theorem. Let B be a complete, atomless, ccc Boolean algebra with t(St(B)) ≤ d(St(B)). If the set {p ∈ St(B) : πχ (x) =
πχ (B)} is dense then πχ (B) = πw(B).
20.13
Torturing J. Greb´ık (bachelor thesis)
The outline of the thesis is as follows:
1. Motivations (Riemann ζ function, square-free numbers, ...)
2. Measure
3. Results
2.1 Density
2.2 P(ω)/Z
Recall definition 19.13. Note that the ideal I generated by sets A such that
lim sup
n→∞
eA (n)
< 1,
eA (n + 1)
is contained in the density zero ideal so, in particular, any ultrafilter extending the dual to Z is not thin.
Take a countable sequence hUn : n < ωi of ultrafilters on ω and take a measure m : P(ω) → [0, 1] extending the
asymptotic density.
40
21
21.1
7. 5. 2014
B. Balcar & P. Simon: Nonhomogeneity of Stone spaces
Consider the additive group of integers (Z, +) and the left-shift f : n 7→ n − 1. We can extend f to the ultrafilters on
Z as follows:
f (U) = {A − 1 : A ∈ U}.
Then the set of nontrivial ultrafilters on the natural numbers N∗ (⊆ Z∗ ) is an f -invariant set. We can define
U − V = V − lim f n (U).
then A ∈ U − V ⇐⇒ {n : A + n ∈ U} ∈ V.
21.2 Question. Can we find an U ∈ N∗ such that U − U = U?
21.3
E. Thuemmel: Yorioka + Zapletal
21.4 Definition (Todorˇcevi´c). Let X be a sequential topological space we define the Todorˇcevi´c ordering T (X) as
follows:
T (X) = {K : K is a union of finitely many converging sequences with limits}
where K ≤ H if H ⊆ K and K 0 ∩ H = H 0 (K 0 is the set of limit points of K).
21.5 Definition. x ∈ 2ω is a random real if it is not contained in any ground-model coded measure zero set.
21.6 Theorem (Yorioka). If T (X) is ccc then T (X) does not add a random real.
Proof. Assuming towards a contradiction assume that some p ∈ T (X) forces that some x˙ ∈ 2ω is random. Let M be a
countable elementary submodel of some (sufficiently large) H(κ) containing everything we need (e.g. p, x,
˙ X, T (X),
...). Now M ∩ 2ω is countable, hence measure zero,Tso there is a sequence of open sets hOn : n < ωi with diameters
going to zero (i.e. µ(On ) → 0) such that M ∩ 2ω ⊆ On . It follows that there is a stronger q ≤ p and n < ω such that
q x˙ 6∈ On . We shall construct a tree. First, for s ∈ 2<ω , let
u(s, x)
˙ = {p ∈ T (X) : p x˙ 6∈ [s]}.
Now, for b ∈ [X]ω , define the tree S(x,
˙ b):
S(x,
˙ b) = {s ∈ 2<ω : u(x) is b-small},
where a set A ⊆ T (X) is b-large if for each b ⊆ c ∈ [X]ω there is q ∈ A such that q 0 ∩ c = b and it is b-small if it is not
b-large. Let b = q 0 ∩ M . Since q is a union of finitely many converging sequences, the set of its limit points q 0 is finite,
so b ∈ M so S(x,
˙ b) ∈ M .
21.7
V Claim. If hAi : i < ni is a sequence of b-large sets then there is a sequence hpi : i < ni with pi ∈ Ai such that
i<n pi 6= ∅.
Proof. By recursion on α < ω1 construct, using the fact that Ai is b-large, a condition pα
i ∈ Ai such that
[
0
(pα
pβi = b.
i ) ∩
β<α,i<n
αi
Let pα = i<n pα
: i < n} is linked
i . Since X is ccc we can use Ramsey’s theorem to find hαi : i < ni such that {p
i
and hence, in our situation, centered. Then also {pα
:
i
<
n}.
i
S
21.8 Claim. The set S(x,
˙ b) is an infinite tree.
Proof. Choose k < ω. We will find s ∈ S(x,
˙ b) of length k. Assume, aiming towards a contradiction, that there is no
such s. Then for each
s
of
length
k
we
have
that u(s) is b-large. By the previous Claim there are ps ∈ u(s) for each
V
s ∈ 2k such that r = s∈2k ps 6= ∅. Then r x˙ 6∈ 2ω — a contradiction.
¨
By Konigs
theorem S has an infinite branch y ∈ M . If y ∈ On then there is s v y such that [s] ⊆ On and (since S is a
tree) s ∈ S. Work in M . By definition of S we know that u(s, x)
˙ is b-small. Let c ∈ [X]ω ∩ M be a witness, i.e. for each
0
r ∈ u(s, x)
˙ we have r ∩ c 6= b. Then, by elementarity, c witnesses that u(s, x)
˙ is b-small in V . Since c is countable we
have c ⊆ M . Letting r = q we arrive at a contradiction since b = q 0 ∩ M .
41
22
22.1
21.5.2014
J. Zapletal: Yorioka’s paper
We aim to show the proof of
22.2 Theorem (Yorioka). Let X be a topological space and T (X) the Todorˇcevi´c ordering. If T (X) is ccc then it does not add
random reals and, in fact, preserves Gδ -coverings.
(this is theorem 21.6).
22.3 Note. There are different definitions of T (X). Here we use definition 21.4.
22.4 Fact. If T (X) is not ccc then there is a condition p ∈ T (X) which forces that ℵ1 is collapsed.
The proof is based on two tricks.
22.4.1
Trick I.
22.5 Definition. Let x ∈ [X]<ω be a finite subset of X. We say that a set Q ⊆ T (X) of conditions is a-large if for each
countable b ⊇ a there is q ∈ Q such that q 0 ∩ b = a, where q 0 is the set of limit points of q.
22.6 Note. If q 0 = a then {q} is a-large.
22.7 Lemma. If T (X) is ccc and hQi : i < ni are all a-large then there are qi ∈ Qi such that
^
qi 6= ∅.
i<n
(see claim 21.7)
Proof. Using the largeness of Qi ’s we can recursively construct a sequence hqiα : α < ω1 , i < ni such that (qiα )0 ∩ qjβ = ∅
for each β < α, j < i.
Then let
pα =
[
qiα .
i<n
Since T (X) is ccc, using an easy inductive argument we can find {αi : i < n} such that {pαi : i < n} is centered
(e.g., for n = 4: choose an uncountable set of compatible pairs and apply ccc to the lower bound of the pairs). Then
it remains to show that
^
qiαi 6= ∅
i<n
22.8 Note. If X is ccc then the set
n_
o
Q : Q is a-big
is a centered system in RO(T (X)).
22.8.1
Trick II.
22.9 Definition. A Gδ -cover of a Polish space X is a system C of Gδ -subsets of X such that for each countable b ⊆ X
there is c ∈ C such that b ⊆ c.
22.10 Lemma. If T (X) is ccc then it preserves Gδ -covers.
Proof. Assume this is not the case and fix a cover C, a name y˙ for a countable subset of 2ω and some condition
p ∈ T (X) that forces that y˙ is not covered by C.
Let M be a countable elementary submodel of Hθ . Since C is a cover, we can find c ∈ C such that M ∩ 2ω ⊆ c. By
our assumption there must be some q ≤ p and n < ω such that q y(n)
˙
6∈ c. Since c is Gδ , there must be some open
o ⊆ 2ω , c ⊆ o and q ≤ p such that q y(n)
˙
6∈ o. Let a = q 0 ∩ M . For m < ω let
Sm = t ∈ 2m : {r : r t 6v y(n)}
˙
is not a-large .
42
Then, by the previous lemma, Sm must be nonempty (otherwise q would force that y(n)
˙
has no initial segment). Note
also that S = {Sm : m < ω} ∈ M . It follows that there is z ∈ M ∩ 2ω a limit point of S. By choice of c we know that
z ∈ c so, in particular, z ∈ o. Since o is open, there is m0 such that [z m0 ] ⊆ o. Now find m > m0 and t ∈ Sm such
that t m0 = z m0 . Now q y(n)
˙
6∈ o so, in particular, q t 6v y(n
˙ ) . Then q ∈ {r : r t v y(n)}
˙
= Q. By a simple
elementarity argument If follows that Q is a-large. This is, of course, a contradiction.
22.11 Corollary. A ccc T (X) cannot add random reals since ground-model null sets form a Gδ -cover which would have to be
preserved.
22.11.1
An Abstract Approach ?
22.12 Axiom. We say that RO(P ) satisfies Axiom Y if for each countable elementary submodel of some Hθ and for
each q ∈ RO(P ) there is a filter F ∈ M such that
{p ∈ M ∩ RO(P ) : q ≤ p} ⊆ F.
22.13 Lemma. If RO(P ) satisfies Axiom Y then it preserves Gδ -covers.
Proof. By contradiction fix a cover C a name {y˙n : n < ω} and a condition p ∈ P such that
p (∀c ∈ C)({y˙n : n < ω} 6⊆ c).
Again fix c ∈ C, an open o ⊇ c and q ≤ p such that q y˙ 6∈ o. Use Axiom Y to find a filter F ∈ M for this q. Define
Sm = {t ∈ 2m : ||t 6v y˙n ||}.
The same argument as before gives us Sm 6= ∅ and the rest of the proof follows as before.
22.14 Observation. If X is ccc then RO(T (X)) satisfies Axiom Y.
22.15
David’s question for Jindra
22.16 Lemma (Forcing Idealized). If P is a forcing such that each ω1 -tree in the extension contains an ω1 -tree in the ground
model then it preserves Suslin Trees.
Proof. Let T0 be a Suslin tree and suppose P forces an uncountable antichain A ⊆ T0 . Let T1 be the tree in the
extension generated by A. This is an ω1 -tree so it contains a ground model ω1 -tree T2 . Let B = {t ∈ T0 : t 6∈
T2 & all initial segments of t are in T2 }. Then B is an antichain and it must be uncountable: otherwise, since T2 is an
ω1 -tree, there must be a t ∈ T2 of height bigger than the height of elements of B. Then we can find an extension of t
in T0 \ T2 and some initial segment of this extension of height bigger than t will be in B — a contradiction.
22.17
J. Zapletal: Iteration of ω-models
22.18 Theorem (Bartoszynski?). If a poset P is sufficiently definable then the statement ”P is ccc” is absolute between generic
extensions.
The original proof uses Keisler’s compactness theorem. We will show a proof using iteration of models.
22.19 Definition. An ω-model M is a countable model of (a fragment of) ZFC with ω M = ω.
22.20 Note. Sufficiently definable in the above theorem means that an ω-model correctly interprets everything that
is needed for the argument.
Proof. Assume that there is an ω-model M such that M |= ”P is not ccc”. Then M is not ccc already in V . To show
this we build a sequence of models hMα : α < ω1 i such that Mα+1 is a generic ultrapower of Mα over P (ω1 )/N S
and a limits we take direct limits. Each Mα will be an ω-model. Finally we take Mω1 . This will be an ω-model which
correctly interprets ”being of size ℵ1 ”. Moreover it will see an uncountable antichain in P (since each Mα sees that)
and this antichain will be uncountable.
43
23
23.1
18.6.2014
J. Greb´ık: Ideals on products of Boolean algebras
23.2 Setting. Let B be a measure algebra and U ∈ ω ∗ . We consider Bω together with four ideals:
1. I0 = {b ∈ Bω :
VW
2. IU = {b ∈ Bω :
V
b = 0}
W
U ∈U
n∈U
bn = 0}
3. I µ = {b ∈ Bω : lim µ(bn ) = 0}
4. IUµ = {b ∈ Bω : U − lim µ(bn ) = 0}
Recall that Bω /IUµ is again some measure algebra.
23.3 Question. When does the equality IU = IUµ hold?
The following can be found in Bartoszynski, Judah: Structure of the real line.
23.4 Definition. A sequence hxn : n < ωi ⊆ B is measure independent if for each finite A ∈ [ω]<ω set of indices we have
^
Y
µ(
xn ) =
µ(xn ).
n∈A
n∈A
23.5 Proposition. Given a sequence or real numbers hai : i < ωi in [0, 1] then there is a measure independent sequence
hxn : n < ωi of elements of B(ω) (the standard measure algebra of length ω) with µ(xn ) = an .
To have
P find a counterexample to question 23.3 we would need a sequence hxn : n < ωi such that U − lim µ(xn ) = 0
and µ(xn ) = ∞. However, we have the following theorem.
23.6 Theorem (Borel). If hxn : n < ωi is measure independent then
X
^ _
µ(xn ) = ∞ ⇒
xn = 1
n<ω
and
X
k<ω n>k
µ(xn ) < ∞ ⇒
n<ω
^ _
xn = 0
k<ω n>k
P
This gives the following conditions on
V U: we
W need that U − lim xn = 0, while n∈U xn = ∞ for each U ∈ U. By the
previous theorem we will also have U ∈U n∈U bn = 1 for each U ∈ U. This gives the following proposition.
23.7 Theorem (Greb´ık). The equality holds IUµ = IU iff U is a semiselective ultrafilter.
where
23.8 Definition (Kunen). An ultrafilter is semiselective iff it is a rapid P-ultrafilter.
The proposition follows from the following characterization of semiselectivity.
23.9 Proposition (Greb´ık). U is semiselective
iff for each sequence han : n < ωi of real numbers in [0, 1] such that U −
P
lim an = 0 there is U ∈ U such that n∈U an < ∞.
23.10 Note (Jech). An ultrafilter U is semiselective iff for eachWBoolean algebra B and each sequence b = hbn : n < ωi
of elements of B such that b ∈ IUµ there is an U ∈ U such that n∈U bn < 1.
23.11
David Chodounsky:
´ Construction of ccc forcing using long diagonalization
The following is motivated by the question whether axiom Y (Y-cc) (see 22.12) is productive.
Recall that
Knaster ⇒ productively ccc ⇒ powerfully ccc ⇒ ccc
and that, consistenly (e.g. under MA), all of the implications can be reversed. Moreover the implications cannot be
separated by ”definable” orderings.
44
23.12 Note (Thuemmel). The Todorˇcevi´c’s ordering, if ccc, is already productively ccc.
23.13 Example. A Suslin tree is ccc but not powerfully ccc.
23.14 Example. Take an indestructible gap T and add a Cohen real c. Then c ∩ T and T \ c are both destructible gaps.
Let P1 and P2 be ccc forcings to, respectively, destroy c ∩ T and T \ c. Then P1 × P2 cannot be ccc, since it destroys T
but T is indestructible. However P1 , P2 are both even powerfully ccc.
23.15 Example (Todorˇcevi´c). If b = ℵ1 then T(R) is productively ccc but not Knaster.
23.16 Example (Baumgartner). The forcing to add an antichain to a Suslin tree by finite conditions is powerfully ccc
but not productively ccc (as shown by the Suslin tree).
We will show that Y-cc does not fit into the above hierarchy. The following example shows that Y-cc is not productive.
23.17 Example (Galvin-Laver). Let X be a space and [X]2 = O0 ∪ O1 . Let
P0 = a ∈ [X]<ω : [a]2 ⊆ O0
ordered by reverse inclusion and
Q0 = [X]<ω
ordered as follows:
a ≤ b ⇐⇒ a ⊇ b & b × (a \ b) ⊆ O0 ,
i.e. a condition is stronger if each new vertex is connected to all old vertices (interpreting (X, O0 ) as a graph).
23.18 Note. If Q0 is ccc then P0 is ccc.
23.19 Proposition. If Q0 is ccc then both Q0 and P0 are Y-cc.
Proof of Claim. So let M be a countable elementary submodel of some Hθ and p ∈ P0 . Let b = p ∩ M and say that a set
S ⊆ P is b-big if for each K ∈ [X \ b]ω there is a a ∈ S such that b ⊆ a and a ∩ K = ∅.
W
Let F = { S : S is b − big}.
23.20 Claim. If p ≤ q ∈ RO(P0 ) ∩ M then q ∈ F.
W
Proof of Claim. Since P0 is dense in RO(P0 ) we have q = {a ∈ P0 : a ≤ q}. To finish the proof of the claim we need
only show that S = {a ∈ P0 : a ≤ q} is b-big. Since S ∈ M we need only check this in M . Let K ∈ M . Since M is
transitive for countable sets, K ⊆ M . Then p ∈ S and p ∩ K ⊆ b. Then, by elementarity, there is p0 ∈ M ∩ S such that
p0 ∩ K ⊆ b.
23.21 Claim. F is centered.
Proof of Claim. We will show that if S0 , . . . , Sn are b-big then there are a0 ∈ S0 , . . . , an ∈ Sn such that
this point, I had to leave the seminar.)
V
i≤n
ai 6= ∅. (At
23.22 Note. It follows from the proof that both Q0 and P0 will be powerfully ccc.
24
24.1
3.9.2014
D. Chodounsky:
´ Generalized Laver Forcing
24.2 Definition. Given a family A ⊆ [ω]ω let L(A) be a forcing consisting of trees T having a stem s such that
(∀t ∈ T )(s ⊆ t → succvT (t) ∈ A).
where succvT (t) = {s(|t|) : s ∈ succT (t)}.
24.3 Note. If A = [ω]ω this is the classical Laver forcing. Laver forcing adds a dominating real. Moreover it has the
Laver property, i.e. each bounded real in the extension is contained in some ground-model tunnel of width 2n .
45
24.4 Note. If A is a filter then L(A) is σ-centered, in general L(A) is just proper (proof by fusion).
24.5 Definition. For T, S ∈ L(A) with the same stem we define T ≤n S if T ∩ ω |stem(T )|+n = S ∩ ω |stem(T )|+n . A
fusion sequence is a sequence of conditions hTn : n < ωi such that Tn+1 ≤n Tn for each n < ω.
24.6 Proposition (fusion). If hTn : n < ωi is a fusion sequence then there is T ∈ L(A) which is a lower bound for all Tn .
24.7 Definition. A family A is hitting if for each X ∈ [ω]ω there is A ∈ A such that |A ∩ X| = ω, it is splitting if
X ∈ [ω]ω there is A ∈ A such that |A ∩ X| = |A \ X| = ω. A family is ω-hitting if for each sequence hXn : n < ωi ⊆ [ω]ω
there is an A ∈ A such that for each n < ω such that |A ∩ Xn | = ω for each n < ω. Similarly we define ω-splitting.
24.8 Note. In the ω-variants it is sufficient to require the intersections to be non-empty.
24.9 Example. Any P-ideal which is hitting is ω-hitting.
24.10 Example. Given a MAD family A the ideal I(A) is hitting but not ω-hitting.
The following theorem is from A. Dow: Two classes of Fr´echet-Urysohn spaces, Proc. Ams. 108(1) 1990.
24.11 Theorem (A.Dow). Laver forcing preserves every ω-hitting and ω-splitting families.
24.12 Definice. A family X is F-omega-hitting if for each sequence hfn : n < ω of functions from ω to ω such that
f [ω] ∈ F there is X ∈ X such that f [X] ∈ F for each n < ω.
24.13 Note. In the above definition, F is usually a co-ideal.
24.14 Observation. If X is F-omega-hitting then it is ω-hitting.
Proof. Let fn be a bijection of Xn onto ω and constant outside Xn .
24.15 Theorem (Hruˇsa´ k, Chodounsky,
´ Guzm´an). The following are equivalent for a filter F and a family X :
1. X is F + -ω-hitting.
2. L(F + ) forces that X is ω-hitting.
Proof of Claim. Suppose 1. fails and let hfn : n < ωi witness this. Let hBn : n < ωi be a partition of ω into infinite
sets (eventually, Bn will be a set of levels of a Laver tree). Let T ∈ L(F + ) be a condition such that for each t ∈ T if
|t| + 1 ∈ Bn then succT (t) ⊆ fn [ω]. Let g˙ be the name for the generic real and let
ˇn \ k]].
X˙ nk = fˇn−1 [g[
˙B
Then T forces that Xnk is infinite. For each X we can find n < ω such that fn [X] ∈ F ∗ . So we prune T to a stronger
S such that for each t ∈ S 0 if stem(S 0 ) ≤ t then succS 0 (t) ∩ fn [X] = ∅. This S forces that X ∩ Xnk is empty for some
k < ω thereby showing that 2. fails.
On the other hand, assume that 1. holds. Let T be a condition and A˙ = hX˙ n : n < ωi a name for a sequence of
infinite sets. Let M be an countable elementary submodel of some sufficiently large H(ϑ) such that T, A ∈ M . Let
hfn : n < ωi enumerate all functions from M with domain ω and range in F + . Use 1. to find X ∈ X such that
fn [X] ∈ F + for each n < ω. We fix this X for the remainder of the proof.
24.16 Claim. Suppose S, A˙ ∈ M such that S A˙ ∈ [ω]ω . Then there is a stronger S 0 ≤0 S such that for each T ≤ S 0
ˇ ∩ A˙ 6= ∅
there is t ∈ T such that St0 ∈ M and St0 X
Proof of Claim. Work in M and, by recursion, construct a sequence htn : n < ωi of nodes of S, a strictly increasing
sequence of natural numbers hkn : n < ωi and a sequence of conditions hRn : n < ωi such that
1. Rn ≤ Stn
2. Rn kn ∈ A˙
3. The set {tn : n < ω} is a maximal antichain in S.
Now let
S0 =
[
Rn
n∈X
We need to show that S 0 ∈ L(F + ) and that it is as required. The second part is obvious, so we show that S 0 is a
condition. For this it suffices to show that for each t ∈ S 0 we have succvS 0 (t) ∈ F + . By 3. there must be n < ω
such that t is compatible with tn . If tn ⊆ t then we are done, as succvS 0 (t) = succvRn (t) ∈ F + . So suppose t ( tn .
Then, again by 3, for each s ∈ succS (t) there is tn(s) compatible with s. Working in M, define f : ω → ω by letting
f (kn(s) ) = s(|t|). Since S is a condition, we have that the range of f is in F + . It follows that succvS 0 (t) = f [X] ∈ F + .
46
We now use the claim to recursively construct a fusion sequence hTn : n < ωi with T0 = T and such that:
ˇ 6= ∅.
1. For each T 0 ≤ Tn there is t ∈ T 0 such that Tn (t) ∈ M and Tn (t) An ∩ X
ˇ hits each A˙ n and this finishes the
The lower bound of the fusion seqence (see 24.6) is a condition which forces that X
proof.
25
10.9.2014
25.1
J. Verner: A simple question
25.2 Definition. Given a topological space X, its extent e(X) is defined to be the smallest cardinality of an open subset
of X.
25.3 Question. Does there exist a space of size (and extent) ℵ1 such that whenever it is partitioned into countably
many pieces, at least one of the pieces contains a subspace of extent ℵ1 .
25.4 Observation. If X is ccc and each of its subspaces has at most countable extent then X is a union of a closed nowhere
dense set and a countable set.
25.5 Corollary. If non(M) = ω1 then any nonmeager subset of R of size ω1 works.
25.6
D. Chodounsky:
´ Axiom Y (joint work with J. Zapletal)
For motivation, recall 22.12.
25.7 Observation. Y -cc is a forcing-property, is preserved on regular subposets and each σ-centered poset has Y .
25.8 Theorem. If P is Y then it is ccc.
For proof see 26.12
25.9 Definition. A forcing P is Y -proper if for each countable elementary submodel of a sufficiently large H(θ) with
P ∈ M and for each p ∈ P ∩ M there is q ≤ p which is a Y -master condition, i.e. satisfies
1. q is a M -generic (i.e. is a master condition in the usual sense)
2. for each r ≤ q there is a filter F ⊆ RO(P ), F ∈ M such that
{s ∈ RO(P ) ∩ M : r ≤ s} ⊆ F.
25.10 Observation. Being Y -proper is a forcing property, is preserved on regular subposets and each Y -cc forcing is Y -proper.
25.11 Proposition. If P is κ-cc, where κ is smaller than the least measurable, then 2. already implies 1. in the above definition.
25.12 Example. The Mathias-Prikry forcing on a measurable cardinal is an example of a forcing satsifying 2. but not
1. in the above definition.
25.13 Question. Does Y -proper + ccc imply Y -cc.
25.14 Theorem. A Y -proper forcing does not add random reals, adds unbounded reals and has many other properties of Y -cc
forcings.
25.15 Example. The forcing to kill all S-spaces, the forcing to force PID, the forcing to show that, under PFA, there
are only 5 Tukey types of orderings, are all Y -proper.
25.16 Example. A forcing which forces (an instance of) OCA cannot be proper (it adds anti-cliques to graphs).
25.16.1
Laver forcings with axiom Y
Recall the definition of L(I + ).
25.17 Theorem. The following are equivalent for analytical P-ideals I:
47
1. The ideal I is an intersection of Fσ -ideals.
2. L(I + ) is Y -proper.
3. For each compact Polish X, open H ⊆ X ω and for each G generic on L(I + ) we have
[
ˇ [Y ]ω ∩ H = ∅ ∃{Yn : n < ω} ∈ V, [Yn ]ω ∩ H = ∅ Yn ⊆
V [G] |= ∀Y ⊆ X,
Yn
n<ω
25.18 Note. If I is an analytical P-ideal then being an intersection of Fσ -ideals is equivalent to being an intersection
of countably many Fσ -ideals. An example of an analytical P-ideal which does not satisfy 1. is the density zero ideal.
Proof of Theorem. 2. → 3.: This is true for all Y -proper forcings P . Let X, H, G, Y˙ be given. Whenever F ⊆ RO(P )
define
n
o
B(F, Y˙ ) = x ∈ X : (∀O ∈ U(x)V [G] )(||O˙ ∩ Y˙ V [G] 6= ∅|| ∈ F)
(note that B(G, Y˙ ) is the closure of Y in V [G] ).
25.19 Claim. B(F, Y˙ ) is an H-anticlique, i.e. [B(F, Y˙ )]ω ∩ H = ∅.
˙
Proof of Claim. Assume, aiming for a contradiction, that there is A = {xn : n < ω}
Q ⊆ B(F, YQ) such that A ⊆ H.
Since H is open, there are open neighbourhoods O0 , . . . , On of x0 , . . . , xn such that i≤n Oi × i≥n X ⊆ H. By the
V
definition of B, we can find pi ∈ F such that pi Oi ∩ Y˙ 6= ∅. Let p = i≤n pi . Then
p
Y
Oi × X ω\n+1 ∩ [Y˙ ]ω 6= ∅
i≤n
However, Y is an H-anticlique, a contradiction (the left side is contained in H).
Aiming for a contradiction, assume that 3. does not hold and that this is forced by some p ∈ P . Let M be a countable
elementary submodel of some sufficiently large H(θ) with P, X, Y, p ∈ M . By Y -properness there is a Y -master
˙
condition
S q ≤ p. Let hYn : n < ωi enumerate all sets of the form B(F, Y ) in M . Then no r ≤ q can force that
˙
x ∈ Y \ n<ω Yn for some x ∈ X ∩ V (since X is second countable) a contradiction.
3. → 1.: Assume 1. fails, we will show that also 3. fails. Since 1. fails, there is A 6∈ I which is in every Fσ
ideal containing I. Let P = L(I + ) and let M be a countable elementary submodel of some sufficiently large H(θ)
containing I, P, .... Let Z = St(RO(P ) ∩ M ). Let X = K(Z) = {K ⊆ Z : K 6= ∅, K compact} with the hyperspace
(Vietoris) topology. Define
\
H = {hKn : n < ωi :
Kn = ∅}.
n<ω
Then H is an open subset of X. Given T ∈ P let
n
o
KT = F ∈ Z : {p ∈ RO(P ) ∩ M : T ≤ p} ⊆ F
then KT ∈ X. Let G be a generic on P and Y = {KT : T ∈ G}. Now Y is an H-anticlique.
Given a sequence
S
hYn : n < ωi of anticliques and T ∈ P it will be sufficient to find S ≤ T such that KS 6∈ n<ω Yn . Since being an
anti-clique is equivalent to being a centered sequence of compact sets (i.e. filters), we may assume that each Yn is an
ultrafilter on RO(P ) ∩ M .
It will be sufficient to find S and a sequence pn ∈ Yn of elements disjoint from S. Since I is an analytic P-ideal, we
can fix a lscsm µ : [ω]<ω → R+
0 with I = Exh(µ).
25.20 Claim. For each k < ω, we can partition A = A0 ∪ · · · ∪ An into pieces in such that for each i < ω we have
|An | = 1 or µ(An ) ≤ 2k .
Proof of Claim. Otherwise, let J be the ideal generated by sets which can be partitioned as in the claim. Then this is
an Fσ -ideal containing I and not containing A, a contradiction.
Note that in the above claim we could have taken the partition to be in M .
25.21 Claim. For any U ∈ P with t = stem(U ) and n < ω there is V ≤0 U (see 24.5) and qi ∈ Yi such that
1. if i < n then qi is disjoint from V ; and
48
2. qi is disjoint from V after deleting finitely many successors of the stem of V .
Proof of Claim. If the condition with stem t and everywhere branching into A is not in Yi then choose a condition
qi ∈ Yi disjoint with this condition and let V =
let B = succvU (t) ∩ A. Then B ∈ I + so limn→∞ µ(B \
PU . Otherwise
−ki
n) = ε > 0. Choose hki : i < ωi such that i<ω 2
< ε/2. Using the previous claim, fix a partition of A into
{Aij : j < nj } with each Aij of submeasure ≤ 2−ki . Since Yn is an ultrafilter and since A ∈ Yn we can find j(i) < nj
such that Aij(i) ∈ Yn . Let
!
[
[
i
0
i
Aj(i) .
B =B\
Aj(i) ∪
i<n
i≥n,j<nj ,|Aij(i) |>1
Finally let V be constructed from U by deleting the successors of t in B 0 . This works.
We now use the claim to construct a fusion sequence Sn (applying the claim to each Stn for |t| = n, j = n). By
proposition 24.5 there is S ≤ Sn . We construct pi as an intersection of a suitable finite set of qi ’s.
This finishes the proof of 3. → 1.
26
24.9.2014
26.1
J. Greb´ık: Graphs & Banach spaces
26.2 Definition. Let P consists of finite graphs with vertices some countable ordinals. Ordered as follows G ≤ H if
V (G) ⊇ V (H) and EG V (H)2 = EH . Let Pα consist of those graphs whose vertices are elements of α.
26.3 Note. Pα is Cohen and is a regular subposet of P which is itself forcing equivalent to adding ω1 many reals.
Also note that P can be written as a two step iteration Pα ∗ Pω\α .
26.4 Proposition. The generic graph added by P does not have an automorphism.
Let us now try a similar forcing in a different setting.
26.5 Definition. A norm || · || : V → R on a space V with basis B is rational if it is generated by finitely many linear
functionals a0 , . . . an : B → Q as follows:
||v|| = max ai (v),
i≤n
for each v ∈ B.
26.6 Note. Each norm can be approximated by a rational norm with the identity being an ε-isomorphism.
26.7 Definition. Let Q consist of finite-dimensional Banach spaces p = (Bp , ap ), where Bp ⊆ ω1 is a finite basis for
the space and || · ||p is a rational norm. A stronger condition must be a larger space and must preserve the norm. The
generic space VG is a union of conditions in the generic. Take WG to be the completion of this space.
26.8 Note. Again, we can define Qα to consist of those conditions whose basis is a subset of α. This will be a regular
subposet of Q which is forcing equivalent to Cohen forcing.
26.9 Proposition. The space WG does not have an automorphism.
26.10
D. Chodounsky:
´ News from Polish
26.10.1
A new class of forcing notions
For motivation, recall 22.12. The following results can be found in D. Chodounsky´ and J. Zapletal: Why Y-c.c.,
submitted.
26.11 Example. Taking completions is necessary in the above definition. Consider collapse of ω1 to ω, i.e. P = {p :
n → ω1 }, then for each elementary submodel M and each p ∈ P we can take F to be all conditions extending ”p ∩ M ”
which is a filter on P containing every condition in M bigger than p. However, P cannot be Y-cc (satisfy axiom Y),
since it collapses ω1 .
26.12 Theorem. σ-centered ⇒ Y-cc ⇒ ccc.
49
Proof. The first implication is easy. Assume for a contradiction P is Y-cc and A ⊆ P is an uncountable antichain. Let
M be a countable elementary submodel of some sufficiently large H(θ). Let A ∈ M and pick q ∈ A \ M . By Y-cc there
is F ∈ M such that {p ∈ RO(P ) ∩ M : q ≤ p} ⊆ F. Let
_
G= B⊆A:
A∈F .
Note that for each B ∈ M we have B ∈ G iff q ∈ B, so G is a non-principal M -ultrafilter on A. Now G is σ-complete
(in M) ultrafilter — a contradiction (ω1 is not a measurable cardinal).
26.13 Example. Forcing specializing an Aronszajn tree is Y-cc
26.14 Example. Forcing freezing an (ω1 , ω1 )-gap is Y-cc.
26.15 Example. The Todorˇcevi´c ordering T (X) is Y-cc if it is ccc. (cf. 21.6).
For basic properties of Y-cc reals, consult theorem 25.14.
26.16 Theorem. Let X be a second countable space and H an open (in the product topology) on X, i.e. an element of [X]ω .
then any anticlique in the extension is covered by a countable set of anticliques from the groundmodel. (cf. 3 of 25.17)
26.17 Theorem. Y-cc forcings preserve ω1 -covers by Gδ -sets of a compact Polish spaces.
26.18 Theorem. A finite support iteration of any length of Y-cc forcings is Y-cc.
Proof. We proof preservation under two-step iteration. So let P be Y-cc and P Q˙ is Y-cc. We need to show that
˙ = {(r, s)
P ∗ Q˙ is Y-cc. So let M be an elementary submodel and (p, q)
˙ ∈ P ∗ Q˙ a condition. Observe that RO(P ∗ Q)
˙ :
˙ We know that there is p0 ≤ p and F˙q ∈ M such that p0 forces that F˙q works for q.
r ∈ RO(P ), r ∈ RO(Q)).
˙ Use Y-cc
for p0 ∈ P to find a filter Fp . Finally let
F = {(r, s) : r ∈ Fp & r s ∈ F˙q }.
The proof for limit stages is a little more complicated.
26.19 Note. If P, Q is Y-cc then P × Q need not be ccc. However it is open whether if P × Q is ccc for P, Q Y-cc forcings
then P × Q is Y-cc.
26.20 Metatheorem. Let ϕ be a property of complete Boolean algebras such that ZFC proves that ϕ implies ccc, is
preserved under iterations and complete subalgebras. Then if 3κ+ (Sκ ) then there is a complete boolean algebra B
having ϕ such that B forces M Aκ (ϕ).
Recall the definition 25.9 of Y-proper forcing notions.
26.21 Example. Laver is Y-proper but not Y-cc.
26.22 Question (Balcar). Is Mathias forcing Y-proper?
The following notion was originally defined in W. Mitchell: Adding closed unbounded subsets of ω2 with finite forcing,
Notre Dame J. Formal Logic 46(3), 2005, 357–371 (definition 2.3).
26.23 Definition (Mitchell). Given a forcing P and a countable elementary submodel M ≺ H(θ) we say that p ∈ P is
a strong master condition if for each r ≤ p there is r0 ∈ M such that for q ∈ M we have r ≤ q → r0 ≤ q.
26.24 Observation. A strong master condition p forces that the generic filter is V -generic on P ∩ M . Cf. this with the fact
that a normal master condition forces the filter to be M -generic.
26.25 Observation. A strongly proper forcing is Y-proper.
26.26 Theorem. Let P be Y-proper, p ∈ P and f˙ a name for a function from κ to κ such that for each countable a ∈ [κ]ω
p f˙ a ∈ V then p f˙ ∈ V .
Proof. Let p, f˙ ∈ M and find a Y-master condition q. Then, by assumption, there is g ∈ V and r ≤ q such that
r f˙ (M ∩ κ) = gˇ. By properness (i.e. since q is a master condition) q f˙[M ] ⊆ M so g[M ] ⊆ M . Y-master gives us
a filter F ∈ M . Given α, β ∈ κ let bα,β = ||f˙(α) = β|| and define h(α) = β iff hα,β ∈ F. Then h is a (partial) function
from κ to κ and is in M .
26.27 Claim. h is a total function.
Moreover h M = g.
50
27
29.10.2014
ˇ arka Stejskalov´a: Consistency of PFA
27.1 S´
For more details one can consult:
• Cummings, J: Iterated Forcing and Elementary embeddings, in Handbook of Set Theory, Springer 2010 (theorem
24.11)
• Devlin, K: Yorkshireman’s guide to proper forcing, in Surveys in Set Theory, CUP 1983
• Jech, T: Multiple forcing, CUP 1986
• Jech, T: Set Theory (3rd edition), Springer 2006
27.2 Definition. A cardinal κ is measurable if there is a normal measure on κ, i.e. a κ-complete ultrafilter extending the
club filter.
27.3 Definition. If U is a normal measure on κ then the mapping:
jU (x) 7→ [cx : κ → V ]U ∈
Y
V /U,
α∈κ
assigning to each set x the (equivalence
class of the) constant function on κ with value x is an elementary embedding
Q
of V into the ultraproduct α∈κ V /U.
27.4 Definition. If j : V → M is an nontrivial elementary embedding then we define cp(j) to be the minimal ordinal
α such that α < j(α).
27.5 Note. If the GCH holds and κ is measurable, as witnessed by U, then κ+ < jU (κ) < κ++
27.6 Definition. A cardinal κ is λ-supercompact if there is an elementary embedding j : V → M with critical point κ
such that
1. M is closed under λ-sequences, i.e. λ M ⊆ M
2. λ < j(κ)
It is supercompact if it is λ-supercompact for each λ.
27.7 Definition. An ultrafilter U on Pκ (λ) = {X ⊆ κ : |X| < κ} is normal if for each hXα : α < λi ⊆ U the diagonal
intersection
\
4α<λ Xα = {y ∈ Pκ (λ) : y ∈
Xα }
α∈y
is an element of U. An ultrafilter is fine if for each α < λ the set {y ∈ Pκ (λ) : α ∈ y} is an element of the ultrafilter.
27.8 Theorem. κ is λ-supercompact iff there is a κ-complete normal fine ultrafilter on Pκ (λ).
27.9 Lemma. Let U be a κ-complete normal fine ultrafilter on Pκ (λ) and j is its associated elementary embeding then
1. [id] = j 00 λ;
2. κ is a critical point of j and λ < j(κ);
3. The range M of j is closed under λ-sequences;
4. j 00 α = [hy ∩ α : y ∈ Pκ (λ)]i for each α ≤ λ, in particular, α = [hotp(y ∩ α) : y ∈ Pκ (λ)i];
5.
λ<κ
M ⊆ M and j 00 Pκ (λ) ∈ M
<κ
6. 2λ
≤ (2λ
<κ
<κ
)M < j(κ) < (2λ
)+
Proof. 1. ⊇: take α < λ, we show that j(α) ∈ [id], i.e. {y ∈ Pκ (λ) : j( α)(y) = id(y)
1. ⊆: Notice that [f ] ∈ [id] ≡ {y ∈ Pκ (λ) : f (y) ∈ y} ∈ U. So we want to find α < λ such that {y ∈ Pκ (λ) : f (y) =
α} ∈ U. Assume, aiming towards a contradiction, that this is not the case. Then for each α < λ the set
Xα = {y ∈ Pκ (λ) : f (y) 6= α}
51
is an element of U. Since U is normal, the diagonal intersection X of these Xα s is in U. But then y ∈ Pκ (λ) : f (y) ∈ y
is not in U — a contradiction.
2.Let α = cp(j) be the minimal ordinal such that [f ] = α < j(α). Then {y ∈ Pκ (λ) : f (y) < cα (y) = α} ∈ U By
κ-completness, if α were below κ, there would be a β < α such that {y ∈ Pκ (λ) : f (y) = β} ∈ U contradicting the fact
that [f ] = α > β. So, κ ≤ cp(j). To show the second part note that λ = otp(j 00 λ) = otp([id]) (by elementarity and 1.).
´ theorem, that
It will be sufficient to show that otp([id]) < j(κ), i.e., by Łos
{y : Pκ (λ) : otp(id(x)) = otp(x) < cκ (x) = κ} ∈ U.
3.This follows directly from the following claim:
27.10 Claim. If j 00 X ∈ M = rng(j) and |Y | ≤ |X| for some Y ⊆ M , then Y ∈ M .
The following theorem was proved by Laver in Laver, R.: Making the supercompactness of κ indestructible under κ-directed
closed forcing, Israel Journal of Mathematics, 29(1978).
27.11 Theorem (Laver). Let κ be a supercompact cardinal. Then there is a function (Laver diamond) f : κ → Vκ such that
for each λ ≥ κ and x ∈ H(λ+ ) there is a supercompact measure U on κ such that jU (f )(κ) = x.
Proof. Let < be a wellordering of Vκ . Define f as follows:
f (α) = min{x : (∃α ≤ γ < κ, x ∈ H(γ + ))(6 ∃U)(jU (f α)(α) = x)} ∪ {∅}
<
This f works. Aiming towards a contradiction, assume not and choose a witness x and λ ≥ κ. Since κ is λλ
supercompact. Let ρ = 22 and j : V → N with ρ < j(κ). Then
N |= (∃ν, x ∈ H(ν + ))(there is no suitable supercompact measure)
ˇ arka did a good job of finishing the proof.)
(Here, I got distracted. Nevertheless, I assume, S´
27.12 Theorem (Baumgartner, 1979). Assume κ is a supercompact cardinal. Then there is a generic extension M with
κ = ω2M and M |= P F A.
Proof of claim. Let f be the laver function given by the previous theorem. We force with a countable support iteration
of length κ: Pκ = hQ˙ α : α < κi such that if f (α) is a Pα -name for a proper forcing, then Q˙ α = f (α) and Q˙ α is trivial
otherwise. Pκ is κ-cc of size κ. Let Q˙ be a Pκ name for the Cohen forcing. By the properties of the Laver function
˙
there is a supercompactness measure U and an embedding jU such that j(f )(κ) = Q.
27.13 Claim. j(P )κ = Pκ .
It follows that Qα = Q˙ for U-many α’s so Pκ adds κ-many subsets of ω. Similarly one can show that κ is collapsed to
ω2 : since this collapse is proper a name for it is small, so we can use the laver function to ”capture it”.
Now let G be a generic filter on Pκ and let Q˙ be a name for a proper forcing in the extension. Find λ ≥ κ such that
λ
˙
Q˙ ∈ H(λ+ ) and find a supercompact measure U on Pκ (22 = ρ) such that jU (f )(κ) = Q.
27.14 Claim. If j : V → M is an elementary embedding with <κ M ⊆ M and if P ∈ M is a κ-cc forcing and G is a
V -generic filter on P then <κ M [G] ⊆ M [G].
Proof of claim. Let s ∈ V [G] be a sequence of length < κ of elements from M [G]. Let s˙ ∈ V be a name for this sequence.
Since the forcing is κ-cc, we can assume s˙ is small so that s˙ ∈ M (by supercompactness of κ).
We want to show that Q˙ is (a name for) a proper forcing in M [G]. The forcing Q is proper in V [G] and this is witnessed
κ
by a club C ⊆ [H(η)]ω of countable elementary submodels of H(η) for some η < 22 . As C ⊆ M [G], by the previous
claim, C ∈ M [G] and it will remain a witness showing that Q is proper also in M [G].
We now continue by showing that, given a sequence hDα : α < ω1 i of dense subsets of Q in M [G] there is a filter in
M [G] meeting all of them.
27.15 Claim (Silver’s lemma). Let j : V → M be an elementary embedding, P ∈ M a forcing and G a V -generic
filter on P . Suppose, moreover, that H is M -generic over j(P ). If j[G] ⊆ H then there is an elementary embedding
j ∗ : V [G] → M [H].
52
Proof of claim. See, e.g., Cummings article referenced above.
Now let g be a V [G] generic filter on Q. We know that jU (P ) = P ∗ Q˙ ∗ R˙ and let H be V [G ∗ g]-generic on R. Then
G ∗ g ∗ H is V -generic on j(P ) and j[G] ⊆ G ∗ g ∗ H so we can use the previous claim to get an elementary embedding
j ∗ : V [G] → M [G ∗ g ∗ H].
Now consider j ∗ (Q) and the sequence hj ∗ (Dα ) : α < ω1 i. If we can show that j ∗ [g] ∈ M [G ∗ g ∗ H], then the filter
generated by j ∗ [g] hits each j ∗ (Dα ), so M [G ∗ g ∗ H] has a generic for these sets so, by elementarity, so does V [G]. So
it remains to show:
27.16 Claim. j ∗ [g] ∈ M [G ∗ g ∗ H]
Proof of claim. It is sufficient to show that j ∗ Q ∈ M [G ∗ g ∗ H]. Since Q˙ is small, we know that j Q˙ ∈ M (since
j Q˙ ⊆ M ). Now
n
o
j Q˙ = hq,
˙ pi, hj(q),
˙ j(p)i : q˙ is a P -name, p ∈ P
Define
τ=
n
o
hq,
˙ j(q)i,
˙ j(p) : q,
˙ is a P -name, p ∈ j(P ) .
It is clear that τ ∈ M and some computation shows that τG∗g∗H = j ∗ Q.
The above claim finishes the proof of the theorem.
28
5.11.2014
ˇ Stejskalov´a: Laver’s diamond (contd.)
28.1 S.
We continued and finished the proof of the consistency of PFA (see theorem 27.12).
29
29.1
19.11.2014
P. Simon: F. Rothberger’s paper: On families of real functions with a denumerable base
The following result is from Rothberger, F: On families of real functions with a denumerable base, Annals of Mathematics,
45 (1944).
29.2 Definition. X has (*) if for each M ⊆ X of size ℵ1 there is a countable S ⊆ X such that each x ∈ M is a limit of
a convergent sequence from S.
29.3 Theorem. The following are equivalent: 1. Rω1 has (); and 2. 2ω1 has () 3. Nω1 has (*)
Proof. ( 2 → 1): Since R ' (0, 1) we will work with the open unit interval. Let M ⊆ (0, 1)ω1 be of size ℵ1 . For each
f ∈ M choose gfi ∈ 2ω1 such that
X
f (α) =
2−i gfi (α).
i<ω
By 2. there is an S ⊆ 2
ω1
ω
countable and Sgfi ∈ S such that Sgfi → gfi . Let
S¯ =
X
|s|
2−i · s(i) : s ∈ S <ω .
i=1
Moreover let
hf,k,n =
k
X
Sgfi (n).
i=1
Clearly hf,k,n ∈ S¯ and
f = lim hf,n,n .
n→∞
The other direction is trivial and other implications are similar.
53
29.4 Theorem. 2ω1 has (*).
Proof. Let M = {fα : α < ω1 } ⊆ 2ω1 . Fix S = ω. We will find an independent system {Aα,i : α < ω1 , i < 2} and
convergent sequences {Sα : α < ω1 } with Sα → fα . We do this by transfinite induction. Let A0,0 , A0,1 be an arbirary
partition of ω into two infinite sets. Moreover let S0 ⊆ ω be an infinite subset of A0,f0 (0) with an infinite complement
in A0,f0 (0) . Suppose we have now constructed Aα = hAβ,i : i < 2, β < αi and Sα = {Sβ : β < α} satisfying the
following:
1. Aα is an independent family;
2. Sα is an almost disjoint family;
T
3. Sβ ⊆∗ γ∈K Aγ,fβ (γ) for each β < α and K ∈ [α]<ω ;
<ω
4. for each
and i : K → 2 there is an infinite B ⊆
T K ∈ [α]
∗
Sβ ⊆
A
.
γ∈K γ,i(γ)
T
γ∈K
Aγ,i(γ) which is almost disjoint with each
For each K ∈ [α]<ω and i : K → 2 choose infinite disjoint subsets
\
MK,i , NK,i ⊆
Aγ,i(γ)
γ∈K
almost disjoint from Sα . By an old result of Bernstein the family {MK,i , NK,i : K ∈ [α]<ω , i : K → 2} has an almost
disjoint refinement by infinite sets so we may as well assume that they are already almost disjoint. Let
Si = {Sβ : β < α & fβ (α) = i},
i < 2.
Now there is a partition of ω into two infinite sets Aα,0 , Aα,1 such that each Sβ ∈ Si is almost contained in Aα,i and
each MK,i ⊆∗ Aα,0 and NK,i ⊆∗ Aα,1 . It remains to construct Sα . The family
{Aγ,fα (γ) : γ ≤ α}
is a countable centered system of infinite sets. Let Sα be an infinite pseudointersection of this family almost dijsoint
from Sβ . This finishes the construction. At the end we let gn (α) = i ↔ n ∈ Aα,i .
29.5
D. Chodounsky: Todorˇcevi´c’s ordering & Y-cc.
29.6 Warning. The following is probably true, but it is not what was actually on the board. The definition of
Todorˇcevi´c’s ordering on the board took finite unions of sequences instead of sets of sequences. And in that case
proposition 29.11 is probably false.
29.7 Definition. Given a topological space Y let Seq(Y ) be the family of convergent sequences in X. The Todorˇcevi´c’s
ordering
on Y , denoted
by T (Y ), consists of finite subsets of Seq(Y ) ordered as follows: p ≤T q if p ⊇ q and
S
S
S
( p)0 ∩ ( q) = ( q)0 .
(See also 21.4).
Recall definition 22.12.
29.8 Theorem (Chodounsky,Zapletal).
T (Y ) is ccc iff it is Y -cc.
´
29.9 Definition. Let X be a set together with a symetric relation π on X. We define the ordering Qπ (X) to consist of
finite subsets of X ordered as follows: p ≤π q if p ⊇ q and p \ q × q ⊆ π.
29.10 Theorem (Yorioka). If T (Y ) is ccc then T (Y ) is Y -cc.
(See also 21.6).
29.11 Proposition. Let X = Seq(Y ) and define π on X by letting (p, q) ∈ π if p and q are compatible in T (Y ). Then Qπ (X)
is forcing equivalent to T (Y ).
Proof. The natural embedding works and is S
even onto the separative quotient of Qπ (X): given any element of the
quotient represented, say, by s. Then write s as a finite union of disjoint convergent sequences. This will be the
preimage of [s].
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30
26.11.2014
30.1
R. Bonnet
30.2 Definition. A Boolean space is a compact zero-dimensional space topological space.
30.3 Definition. A meet semilattice is a pair hS, ∧i where ∧ is a symmetric associative operation on S. If the universe
S is a topological space, we also assume ∧ to be continuous.
30.4 Note. Any linear order of a space X with a minimal element induces a (meet) semilattice operation by considering x ∧ y = inf {x, y}.
30.5 Example. The Cantor set 2ω is linearly ordered by the lexicographic order and thus gives rise to a meet semilattice.
30.6 Definition. (X, ≤) is a Priestley space if it is a Boolean space with a partial order such that whenever y 6≥ x then
there is a clopen ≤-final subset of X containing x but not y.
30.7 Note. It follows from the definition that the partial order must be a closed relation in X.
30.8 Example. A Boolean space with equality is a Priestley space.
30.9 Definition. Let (X, ≤) be a Priestley space. The hyperspace of X, denoted by H(X), consists of all nonempty
compact final subsets of X. Given a clopen U ∈ H(X) let U + = {K ∈ H(X) : K ⊆ U }. The
{U + , H(X) \ U + : U clopen & U ∈ H(X)}
is a basis of a zero-dimensional topology on H(X).
30.10 Note. If the order is equality, then the topology of H(X) coincides with the Vietoris topology. This is also the
case if H(X) is closed in the standard hyperspace. In general, however, it is not clear whether the Vietoris topology
cannot be stronger.
30.11 Example. A Boolean LOTS X is a Priestley space and H(X) is homeomorphic to X.
30.12 Proposition. The space H(X) is compact, Hausdorff.
30.13 Proposition. There is a natural continuous semilattice operation on H(X): given A, B ∈ H(X), we let A ∧ B = A ∪ B.
30.14 Theorem. For each Priestley space X and each continuous increasing function f : X → Y into some Boolean semilattice
there is a unique continuous, ∧-preserving ”lifting” F : H(X) → Y such that f (x) = F (i(x)), where i(x) = [x, →) is the
natural embedding of X into H(X).
30.15 Corollary. Any continuous increasing map ϕ between two Priestley spaces can be uniquely and naturally extended to a
map ϕˆ between their hyperspaces.
30.16 Definition. A Priestley space X is a Skula space if each closed final segment is clopen.
30.17 Definition. A Boolean algebra B (or its Stone space S(B)) has a clopen selector if there is a family U = {Ux : x ∈
S(B)} ⊆ B such that
1. x ∈ Ux for each x ∈ S(B);
2. x ∈ Uy → Ux ⊆ Uy for each x ∈ S(B); and
3. for each x 6= y either x 6∈ Uy or y 6∈ Ux
30.18 Example. The one-point compactification αD of a discrete space has a clopen selector: Ud = {d} and Uα = D.
30.19 Proposition. If S(B) has a clopen selector then it is scattered.
30.20 Example. There is a scattered space (even one with Cantor-Bendixon rank 3) which does not have a clopen
selector (or, equivalently, which is not a Skula space).
Proof. Fix A = {Aα : α < λ} an almost disjoint family on ω and {fα : α < λ} a dominating family of functions from
ω to ω. Consider
X = ω × (ω + 1) ∪ A ∪ {∞}
with each {n} × ω converging to (n, ω). Moreover, each (n, m) ∈ ω × ω is clopen. Then
gα = Aα × (ω + 1) \ graph(fα ) ∪ {Aα }
is a neighbourhood of Aα .
Assume that it has a clopen selector. We may assume that each member of the selector does not contain ∞. Applying
the Delta system lemma we continue to work a bit and arrive at a contradiction.
55
30.21 Theorem. S(B) has a clopen selector iff B is generated (as a lattice) by a well-founded set iff it is a Skula space.
30.22 Proposition. If X is Skula then so is H(X).
30.23 Proposition. cb(X) ≤ wf (H(X)) ≤ ω cb(X) .
´
30.24 Question. Can we find an almost disjoint family A of size ℵ1 such that the Mrowka
space ψ(A) does not have
a continuous semilattice operation?
31
31.1
3.12.2014
J. Greb´ık: Questions around Suslin trees
31.2 Definition. Given a tree T and a coloring χ : T → 2 we say that a level α returns to β < α if there is a color c ∈ 2
and an increasing seqence hβn : n < ωi cofinal in α such that for each t ∈ Tβ the set {n : χ(t βn ) 6= c} is finite.
31.3 Proposition. If for each coloring χ : T → 2 there is a level β and an increasing sequence hαδ : δ < ω1 of levels returning
to β then RO(T ) with the sequential topology is compact.
32
32.1
7.1.2015
Michael Hruˇsa´ k: Weak Dimond
32.2 Definition (Jensen). A 3-sequence on ω1 is a sequence hAα : α < ω1 i of subsets of ω1 such that Aα ⊆ α and
for each X ⊆ ω1 the set {α < ω1 : X ∩ α = Aα } is stationary. The diamond principle is the statement that there is a
3-sequence.
32.3 Definition (Devlin-Shelah). The weak diamond is the statement that for each coloring F : 2<ω1 → 2 there is a
diamond sequence g : ω1 → 2 such that for each branch f ∈ 2ω1 the set
{α < ω1 : g(α) = F (f α)}
is stationary.
32.4 Note. Weak diamond is equivalent to 2ω < 2<ω1 .
32.5 Definition. Given sets A, B and a relation E ⊆ A × B such that E ⊆ A × B satisfies A ⊆ E −1 [B] and A 6⊆ E −1 (b)
for each b ∈ B, define
hA, B, Ei = min{|B 0 | : B 0 ⊆ B & (∀a ∈ A)(∃b ∈ B 0 )aEb}
The following definition comes from Dˇzamonja, M., Hruˇsa´ k, M. and Moore, J.: Parametrized 3 principles, Transactions
of AMS, 356.6 (2004).
32.6 Definition. A cardinal invariant is Borel provided it can be written in the form hA, B, Ei for some Borel A, B, E.
32.7 Definition. The parametrized diamond Φ(A, B, C) is the statement that for each coloring F : 2<ω1 → A there is a
guessing sequence g : ω1 → B such that for each branch f ∈ 2ω1 the set
{α < ω1 : (F (f α), g(α)) ∈ E}
is stationary. The parametrized weak diamond 3(A, B, E) is the same statement where we restrict it to ”definable
colorings”.
32.8 Note. Typically, a coloring is definable means that each of its restrictions to each 2α is Borel.
32.9 Definition. The strongest weak diamond (denoted Φ(ω1 ) )is the principle Φ(ω1 , ω1 , =).
32.10 Theorem. The (standard) 3 principle is equivalent to CH plus the strongest weak diamond.
32.11 Theorem. The strongest weak diamond and hA, B, Ei ≤ ω1 implies Φ(A, B, E).
32.12 Exercise. The strongest weak diamond does not imply CH.
Proof. Start with a model of GCH, add ℵ2 -many Cohen reals followed by F n(ω3 , 2, ω1 ).
56
Unfortunately, the strongest weak diamond implies 2ω < 2ω1 .
32.13 Example. The strongest weak diamond + non(M) = ℵ1 implies the existence of a Suslin tree.
32.14 Definition. The diamond principle 3(ω1 , ω1 , =), abbreviated as 3(ω1 ), is Φ(ω1 , ω1 , =) restricted to colorings
from L(P(ω1 ))
Most of the things true for Φ(ω1 ) are also true even for 3(ω1 )).
32.15 Example. 3(ω1 ) + non(M) = ℵ1 implies the existence of a Suslin tree.
32.16 Theorem. 3(ω1 ) plus CH is equivalent to 3.
32.17 Theorem. If a forcing P is ”sufficiently definable”, proper of size ≤ c then the countable support iteration Pω1 forces
3(ω1 ).
32.17.1
Generalizations ?
32.18 Definition. 3(ω2 ) is the statement that for each coloring F : 2<ω2 → ω2 in L(P(ω2 )) there is a guessing sequence
g : ω2 → ω2 such that for each branch f : ω2 → 2 such that the set
{α : F (f α) = g(α)}
is stationary.
32.19 Conjecture (Hruˇsa´ k). 3(ω2 ) is a theorem of ZFC.
Recall Baumgartner’s result that, under PFA, all ℵ1 -dense sets of reals are order isomorphic. This motivates the
question:
32.20 Question. Is it consistent that all ℵ2 -dense sets of reals are order isomorphic.
32.21 Theorem. Devlin-Shelah’s weak diamond restricted to colorings from L(P(ω1 )) implies that there are two non-isomorphic
ℵ1 -dense subsets of R.
Proof of Theorem. Fix some set X = {xα : α < ω1 } ⊆ R and, without loss of generality, assume that Q ⊆ X. By
suitable coding (using the first ω levels of 2<ω1 ), we may assume that each s ∈ 2<ω1 codes an order embedding hs of
Q into R and some xsα . Each hs can be uniquely extended to an order isomorphism hs : R ,→ R. Define
F (s) = 0 ⇐⇒ hs (xsα ) ∈ X.
Clearly F is in L(X). Now, using the weak diamond, let Y = {α : g(α) = 1}. Then Y clearly cannot be order
isomorphic to X. Y need not be ℵ1 dense, but this can be mended in the obvious way by, e.g., using odd levels to
guarantee this and even levels as we defined F above.
32.22 Note. The proof works also for ℵ2 . In particular, 3(ω1 ) implies this. This means that if 3(ω1 ) is a theorem of
ZFC then Baumgartner’s result could not be extended to ℵ2 .
32.22.1
Applications in topology
Recall that
32.23 Definition. A topological space X is sequential if for each A ⊆ X which is not closed there is a converging
sequence hxn : n < ωi in A whose limit is not in A. The order ord(X) of a sequential space is the smallest ordinal α
such that iterating sequential closure α-many times always gives the full closure.
32.24 Theorem (Baˇstirov,70s). CH implies that there is a compact sequential space of order ω1 .
32.25 Theorem. There is a compact sequential space of order 2 (e.g. a one-point compactification of a Ψ-space). If we assume
b = c, Dow showed that we can get one of order 4. Under MA, such a space of order 5 exists.
´
There is also Balogh’s solution to the Moore-Mrowka
problem:
32.26 Theorem (Balogh,80s). Under PFA every compact countably tight space is sequential.
32.27 Definition. A space X is super-sequential if it is sequential, scattered and the CB-rank coincides with the sequential order.
57
32.28 Theorem (Dow,Shelah). Under PFA each compact super sequential space has sequential order at most ω.
32.29 Theorem (Hruˇsa´ k). 3(ω ω , ω ω , 6≥∗ ) implies that there is a supersequential compact space X with order ω1 .
32.30 Note. If a space X is locally countable than any order 2 sequential space contains a Ψ-space.
32.31 Theorem (Hruˇsa´ k). 3(ω ω , ω ω , 6≥∗ ) (also abbreviated 3(b)) implies that there is a supersequential compact space X
with order 3.
Proof. We need a MAD A family on ω and a family B of infinite subsets of ω such that
1. for each B ∈ B, A ∈ A either A ⊆∗ B or A ∩ B =∗ ∅;
2. B is I(A)-almost disjoint.
3. for each sequence hAn : n < ωi there is a B ∈ B almost containing infinitely many An ’s
Once we have them, we take the space Ψ(A) add B on top with the natural topology and do a one-point compactification (formally, take the Stone space of the subalgebra of P(ω) generated by A ∪ B ∪ ω). This space will have sequential
order 3. To use the diamond we use coding to associate with each s ∈ 2<ω1 a triple (Xs , hBβs : β < αs i, hAsβ : β < αs i).
Moreover, fix a borel function ϕ which transforms this triple so that the Bs form a partition of ω and As are partitions
of the Bs. If Xs ⊆ ω, then F (s) is the function ds = max Asn ∩ Xs for Asn s partitioning either all B s s or the least B s
which hits Xs infinitely often (this will guarantee that A and B are MAD). If Xs is a sequence of infinite sets, we
similarly guarantee 3. Then, recursively construct B, A at each step using the guessing function g to go to the next
step.
32.32 Note. A similar proof shows that 3(s) gives a counterexample to the Scarborough-Stone problem.
58