book of abstracts

ICTM2015
International Conference on Topology, Messina
September 7 - 11, 2015
http://mat521.unime.it/ictm2015
Villa Pace, Messina University
On the occasion of Filippo Cammaroto’s 65th Birthday
BOOK OF ABSTRACTS
PRIMAL SPACES AND QUASIHOMEOMORPHISMS
HAOUATI AFEF AND SAMI LAZAAR?
Abstract. In his paper ”the categories of flows of Set and Top”, O.Echi has
introduced the notion of primal spaces.
The present paper is devoted to shedding some light on relations between
quasihomeomorphisms and primal spaces.
Given a quasihomeomorphism q : X → Y , where X and Y are principal
spaces, we are concerned specifically with a main problem: what additional
conditions have to be imposed on q in order to render X (resp.Y ) primal when
Y (resp.X) is primal.
(Haouati Afef) Department of Mathematics, Faculty of Sciences of Tunis. University Tunis-El Manar. “Campus Universitaire” 2092 Tunis, TUNISIA.
E-mail address, Haouati Afef: haouati.afef@yahoo.fr
(Sami Lazaar) Department of Mathematics, Faculty of Sciences of Tunis. University
Tunis-El Manar. “Campus Universitaire” 2092 Tunis, TUNISIA.
E-mail address, Sami Lazaar: salazaar72@yahoo.fr
2000 Mathematics Subject Classification. 54B30, 54D10, 54F65, 54H20.
Key words and phrases. Quasihomeomorphism; Principal space; Sober space.
? Corresponding author.
1
Finite Unions of “nice” Subspaces
Arhangel’skii A.V.
MPGU and MGU, Moscow, Russian Federation
Abstract
All spaces under consideration are assumed to be Tychonoff topological spaces. We discuss how much we know about the structure
of topological spaces X which can be represented as the union of
a not too large collection of subspaces with a certain nice property.
In particular, we consider finite unions of subspaces with the weight
≤ τ , finite unions of subspaces with a point-countable base, and finite
unions of metrizable subspaces. As a corollary of this approach, the
classical A.S. Mischenko’s Theorem on metrizability of compacta with
a point-countable base [10] has been extended in [1] to finite unions
in the following way:
Theorem 0.1. Suppose that X is a countably compact space such
that X = ∪{Mi : i = 1, ..., n}, where each Mi is a space with a pointcountable base. Then X = ∪{Fi : i = 1, ..., n}, where each Fi is
closed in X, Fi ⊂ Fi+1 for i = 1, ..., n − 1, and each of the spaces
F1 , F2 \ F1 , ..., Fn \ Fn−1 is metrizable and locally compact.
Of course, the finite unions of metrizable subspaces are of special
interest. See in this connection [5], [6], [11], [7], [8]. For example, M.E.
Rudin has shown that every compact space which is the union of two
metrizable subspaces is an Eberlein compactum (see [9]). Notice that
a compactum which is the union of three metrizable subspaces needn’t
be an Eberlein compactum or a Corson compactum.
Recently, it has been shown that the case of finite unions of dense
metrizable subspaces deserves particular attention [2], [1], [3]. Again,
spaces of this kind needn’t be metrizable. However, all of them are
easily seen to have a σ-disjoint base. Hence, if a compact space X is
the union of a finite family of dense metrizable subspaces, then X is
metrizable, by Mischenko’s Theorem. However, Mischenko’s Theorem
1
does not generalize to Lindel´’of spaces: there exists a non-metrizable
Lindelöf space with a σ-disjoint base, - just take the version of the
Michael line generated by a Bernstein subset of the space of real numbers. This space is the union of two metrizable subspaces. However,
it is less easy to answer the question posed by M.V. Matveev: does
there exist a non-metrizable Lindelöf space X such that X = Y ∪ Z,
where Y and Z are dense metrizable subspaces of X?
A space X is pseudo-ω1 -compact, if for every uncountable family
ξ = {Uα : α ∈ A} of nonempty open subsets of X there exists x ∈ X
such that every neighbourhood of x intersects Uα for infinitely many
α ∈ A. Clearly, every Lindelöf space is pseudo-ω1 -compact. Every
pseudocompact space is also pseudo-ω1 -compact.
Theorem 0.2. Suppose that a pseudo-ω1 -compact space X is the
union of a finite family µ of dense metrizable subspaces of X. Then
X is separable and metrizable.
Corollary 0.3. If a Lindel´’of space X is the union of a finite family
µ of dense metrizable subspaces, then X is separable and metrizable.
Corollary 0.4. If a pseudocompact space X is the union of a finite family µ of dense metrizable subspaces, then X is separable and
metrizable.
The last three statements have been recently obtained in [2].
See [4][Theorem 2.15] for yet another result of similar nature on
dense unions.
Problem 0.5. Must a space X be Dieudonné complete if it can be
represented as the union of two (of finitely many) dense metrizable
subspaces?
This question has been posed in [2].
References
[1] A.V. Arhangel’skii,Structure theorems for finite unions of subspaces of
special kind. Submitted (May 2015).
[2] A.V. Arhangel’skii, Addition theorems for dense subspaces. Submitted
(May 2015).
2
[3] A.V. Arhangel’skii, M.M. Choban, Dense subspaces and addition theorems for paracompactness. Topology and Appl. 185-186 (2015), 23-32.
[4] A.V. Arhangel’skii and S. Tokgöz, Paracompactness and remainders:
around Henriksen-Isbell’s Theorem. Q and A in General Topology 32
(2014), 5-15.
[5] J. Chaber, Locally finite unions of metric spaces. Mat. Japon. 26:3
(1981), 271 - 274.
[6] G. Gruenhage, Metrizable Spaces and Generalizations. In: M. Husek and
J. van Mill, Eds, Recent Progress in General Topology, 2. North-Holland,
Amsterdam, 2002, Chapter 8, 203 - 221.
[7] M. Ismail, A. Szymanski, On the metrizability number and related invariants of spaces. Topology and its Appl. 63 (1995) 6977.
[8] M. Ismail, A. Szymanski, On the metrizability number and related invariants of spaces, II. Topology and its Appl. 71 (1996) 179191.
[9] E.A. Michael and M.E. Rudin, Another note on Eberlein compacta. Pacific J. Math. 72:2 (1977), 497–499.
[10] A. Mischenko, Spaces with a point-countable base, Soviet Math. Dokl.
3(1962), 855-858.
[11] S. Oka, Dimension of finite unions of metric spaces. Math. Japon. 24
(1979), 351 –362.
e-mail address: arhangel.alex@gmail.com
3
PDO calculus on noncompact manifolds
equipped with the cocompact action of
groups
Andronick A. Arutyunov
Moscow Institute of Physics and Technology
Andronick.Arutyunov@gmail.com
This report is dedicated to get the theory of differentional and pseudodifferentional
operators, which act on a noncompact manifold.
The main idea is to research an isomorphism whict reduce functions on
some types of noncompact manifolds to elements of space which is equivallent
to the smooth sections of stratification. The stratification is received as a
factor-space of cocompact action of the group.
Λ : f (z) →
X
γf (γz)
γ∈Γ
So, we get a function which acts from compact factor-manifold to groupalgebra.
The main idea of the following construction is to find all derivations –
linear operators which consider to
D(f g) = D(f )g + f D(g).
And to define the order of such operators as a composite order as a РґРӨРҸР№Р¤РғРђРӕ
of number of derivations and the length of commutator in the coefficent. For
1
example the operator A, given by formula
A = [U, V ]D,
where D – is a derivation and U , V – is a quaziperiodic functions is an
operator of zero-order. Intresting example is acting of Fouks-groups on
hyperbolic spaces and other manifolds where noncompact groups act. Classificitators:
2
[35S05], [35S99].
3
A game for Gδ -diagonal spaces.
Leandro Aurichi
Department of Mathematics, Sao Paulo University, Brazil
aurichi@icmc.usp.br
Based on a result of Ceder, we present a selection principle that characterizes
spaces which have the Gδ -diagonal property. Then we work with the related
game and obtain a broader class. Some results and examples are also given.
This is a joint work with D. A. Lara.
1
Intersection of a set with a hyperplane
M. V. Balashov
Department of Higher Mathematics, Moscow Institute of
Physics and Technology (state university)
email: balashov73@mail.ru
Abstract
In the present report we consider the set-valued mapping whose
images are intersections of a fixed closed convex bounded set A with
nonempty interior from a real Hilbert space with shifts of a closed
linear subspace L:
\
A 3 x → F (x) = A (L + x) .
We characterize such strictly convex sets in the Hilbert space, that
the considered set-valued mapping F is Hölder continuous with the
power 12 in the Hausdorff metric. We also consider the question about
intersections of a fixed uniformly convex set [1] with shifts of a closed
linear subspace. We prove that the modulus of continuity of the setvalued mapping in this case is the inverse function to the modulus of
uniform convexity [2, Theorem 3.1] and vise versa: the modulus of
uniform convexity of the set is the inverse function to the modulus of
continuity of the set-values mapping.
MSC 2010. Primary: 49J52, 46C05, 26B25. Secondary: 46B20,
52A07.
References
[1] B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math, 7,
(1966), 72 — 75.
[2] M. V. Balashov, D. Repovš, Uniform convexity and the spliting problem
for selections, J. Math. Anal. Appl. 360:1 (2009), 307-316.
1
On some generalizations of cardinal inequalities
for Hausdorff and Urysohn spaces
Andrei Catalioto
Department of Mathematics and Computer Science – University of Messina
email: acatalioto@gmail.com
Abstract
Here are presentend some historical well-know cardinal bounds for
Hausdorff and Urysohn spaces and their relative generalizations and
variations. This represent a survey on two articles wrote by the author
jointed with F. Cammaroto and J. Porter.
Following the best-known Arhangel’skiı̆’s inequality for Hausdorff spaces
|X| ≤ 2L(X)χ(X) (1969 - [1]), it is showed that |X| ≤ 2aLψc (X) (2013 [3]). Similarly, following the best-known Bella-Cammaroto’s inequality for Urysohn spaces |X| ≤ 2aL(X)χ(X) (1988 - [2]), it is proved that
|X| ≤ 2aLψ(X) (2013 - [4]).
References
[1] A.V. Arhangel’skiı̆, On the cardinality of bicompacta satisfying the first
axiom of countability, Soviet Math. Dokl. 10 (1969), 951-955.
[2] A. Bella and F. Cammaroto, On the cardinality of Urysohn spaces,
Canad. Math. Bull. 31 (1988), 153-158.
[3] F. Cammaroto, A. Catalioto and J. Porter, On the cardinality of Hausdorff spaces, Topology Appl. 160 (2013), 137-142.
[4] F. Cammaroto, A. Catalioto and J. Porter, On the cardinality of Urysohn
spaces, Topology Appl. 160 (2013), 1862-1869.
1
Some properties of the Lie algebra of a topological
abelian group
Marı́a Jesús Chasco,
Department of Physics and Applied Mathematics
email: mjchasco@unav.es
The author acknowledges the financial support of the Spanish Ministerio de
Economı́a y Competitividad grant MTM 2013-42486-P
Abstract
For an abelian topological group G, The vector space CHom(R, G)
endowed with the compact open topology is called the Lie algebra of
the topological group G and denoted by L(G) in analogy with the
classical theory of Lie groups. In that case the evaluation mapping
L(G) −→ G, φ 7−→ φ(1)
is continuous and it is called the exponential function expG . The elements of im expG are those lying on one–parameter subgroups, and
G is the union of its one–parameter subgroups if and only if expG is
onto. It is well known that this happens for example when G is a locally
compact arc–connected abelian group. We find new classes of abelian
groups for which the corestriction of expG into the arc–component of
the group G is surjective and open.
References
[1] L. Außenhofer, M. J. Chasco, X. Domı́nguez. Arcs in the Pontryagin dual of a
topological abelian group, J. Math. Anal. Appl. 425, Issue 1, 337–348, (2015).
[2] M. J. Chasco. One-parameter subgroups of topological abelian groups, Topol.
Appl. 186, 33-40, (2015).
1
Note on Extended Bitopological Ultra Ri -Spaces
1
M.Lellis Thivagar
2
M.Davamani Christober
3
V.Ramesh
1,3 School
of Mathematics, Madurai Kamaraj University
Madurai - 625 021, Tamil Nadu, INDIA
1 E-mail : mlthivagar@yahoo.co.in
3 E-mail : kabilanchelian@gmail.com
2 Department of Mathematics, The American College
Madurai - 625 002, Tamil Nadu, INDIA
E-mail : christober.md@gmail.com
Abstract :
In 1963, Kelly initiated the study of the bitopological space which is to be a set X
equipped with two topologies τ1 and τ2 on X. Recently new bitopological notions of τ1,2 open sets and τ1,2 -closed sets have been introduced . Further the extended bitopological
space is initiated and characterized their properties by Lellis Thivagar et al. Also in
1974, Dube et al introduced some more separation axioms RY , RY S and RD , which
are weaker than R0 . . In this paper, we introduce some new separation axioms by using
(1, 2)∗ -α-open sets, (1, 2)∗ -semi-open sets, (1, 2)∗ -pre-open sets in extended bitopological ultra spaces called as Ultra+ -Ri , Ultra+ semi-Ri (i= 0,1) Ultra+ -RD , Ultra+ -RT and
Ultra+ -RY S -spaces and we develop some more weak separation axioms of R0 . Further
we derive its various properties and relation between other existing spaces. The most of
the results in this paper can be extended to Digital Topology.
Keywords : Ultra+ -Ri , Ultra+ semi-Ri (i= 0,1) , Ultra+ -RD , Ultra+ -RT ,
Ultra+ -RY S -spaces.
2010 AMS Subject Classification : 54C55, 54D10.
References
[1] Kelly.J.C.:Bitopological spaces, Proc.London Math. Soc(3), 13(1963), 71-89.
[2] Lellis Thivagar.M, Arockiadasan.M, Jayaparthasarathy.G.:Remarks On Weakly
Open sets in Extended Bitopological Spaces , International Research Journal of
Mathematical Sciences, Vol.3(2014),714-717 .
[3] Lellis Thivagar.M, Raja Rajeswari.R and Athisaya Ponmani.S.: Characterizations
of ultra-separation axioms via (1, 2)α-kernel, Lobachevski Journal of Mathematics,Vol 25 ( 2005),50-55.
[4] Misra.D.N and Dube.K.K .: Some axioms weaker than the R0 axiom, Glasnick Mat.
ser III,8,(1973), 145-147.
1
On topological groups with remainder of
character k
Maria Vittoria Cuzzupé, University of Messina, Italy
Abstract
We establish that the character of a non-locally topological group,
which has a remainder of character κ, does not exeed κ+ . This represents a generalization of a result given in A.V. Arhangel’skii and
J.van Mill, On topological groups with a first-countable remainder,
Top. Proc. 42 (2013), 157–163. We also show that this estimate
is the best possible by constructing a non-metrizable non locally compact topological group with a remainder of character κ.
Coauthor: Maddalena Bonanzinga, University of Messina, Italy
1
IDEAL QUASI-NORMAL CONVERGENCE AND RELATED
NOTIONS
Pratulananda Das
Department of Mathematics, Jadavpur University
Kolkata-700032, West Bengal, India
email: pratulananda@yahoo.co.in
(This is joint work with Prof. Lev Bukovský and Dr. Jaroslav Supina, UPJS
University, Kosice, Slovakia)
abstract: Recently the author and D. Chandra began to study the notion of an
ideal quasi-normal convergence and some topological notions defined by this convergence. We show that several properties of so introduced notions depend on the
ideal and sometimes, they are also equivalent to some important property of the
ideal. Moreover, we show non-trivial cases when the new notion introduced by the
ideal quasi-normal convergence is equivalent to the corresponding original notion.
Some relations between introduced notions for different ideals are investigated as
well. We also investigate certain characterizations involving selection principles of
function spaces and also certain types of open covers.
Keywords: Ideal, pseudounion, I-quasi normal conergence, IQN-space, IwQNspace, Arkhangel’skii’s properties.
Mathematical Reviews subject classification: 40A35, 54G15.
References
[1] L. Bukovský, I. Reclaw, M. Repick, Spaces not distinguishing pointwise and quasi-normal
convergence of real functions, Topology Appl. 41 (1991), 25 - 40.
[2] L. Bukovský, I. Reclaw, M. Repick, Spaces not distinguishing convergence of real valued
functions, Topology Appl. 112 (2001), 13 - 40.
[3] L. Bukovský, J. Hale, QN-spaces, wQN-spaces and covering properties, Topology Appl.154
(2007), 848 - 858.
[4] P. Das, Certain types of open covers and selection principles using ideals, Houston J. Math.
39 (2013), 637 - 650.
[5] P. Das, D. Chandra, Spaces not distinguishing pointwise and I-quasinormal convergence of
real functions, Comment Math. Univ. Carolin. 54 (2013), 83 - 96.
[6] G. Di Maio and Lj. D. R. Kočinac, Statistical convergence in topology, Topology Appl. 156
(2008), 28-45.
[7] R. Filipw, M. Staniszewski, On ideal equal convergence, Cent. Eur. J. Math. 12 (2014), 896
- 910.
1
Point-picking games with bounded finite selections
Leandro F. Aurichi, Angelo Bella and Rodrigo R. Dias
Instituto de Matemática e Estatı́stica, Universidade de São Paulo
roquedias@usp.br
Abstract
Motivated by the work [2], in which productivity of countable tightness is related to countable strong fan tightness and its game version
G1 (Ωp , Ωp ), we study variations of this game in which the second player
is allowed to pick, in each inning, a finite number of points that has
been fixed in advance. If this finite number is k > 0, we denote the
corresponding game by Gk (Ωp , Ωp ).
Our main result is that, for each k ∈ N, the games Gk (Ωp , Ωp ) and
Gk+1 (Ωp , Ωp ) are distinct — although all of the selective properties
Sk (Ωp , Ωp ) for k ∈ N are equivalent. We also investigate how these
games relate to other variations with finitely many selections, such as
Gf (Ωp , Ωp ) — in which the second player may pick f (n) points in the
n-th inning, for some f ∈ ω ω fixed in advance — and Gfin (Ωp , Ωp ).
References
[1] A. V. Arhangel’skiı̆. The frequency spectrum of a topological space and
the product operation. Trudy Moskovskogo Matematicheskogo Obshchestva, 40, 171–206, 1979.
[2] L. F. Aurichi and A. Bella. Topological games and productively countably tight spaces. Topology and its Applications, 171, 7–14, 2014.
[3] M. Scheepers. Combinatorics of open covers (III): games, Cp (X). Fundamenta Mathematicae, 152, 231–254, 1997.
1
ALMOST AUTOMORPHIC DERIVATIVE OF AN
ALMOST AUTOMORPHIC FUNCTION
S. D. DIMITROVA-BURLAYENKO
Here we will use the following notations. R is a set of all real numbers,
Y is a Banach space, f (t) is an abstract function from R into Y, and
f 0 (t) is a derivative of f (t). An function f (t) is a compact, if the range
of f (t) is relatively compact in Y. fs (t) = fs (t + s) is a translation,
when s is a number, while in the case a = an is a sequence, then fa (t)
is a pointwise limit (if the latter exists) of the sequence f (t + an ).
It is well known that a uniform continuity of the derivative of an
almost automorphic (almost periodic) function [2] guarantees the a.a.
(a.p.) derivative. Author reduces the requirement of the uniform continuity of the derivative, [3]. In this paper we propose other requirements, which ensure the a.a.(a.p.) derivative. The following basic
theorems are formulated.
Theorem 1. Let function f (t) be a.a. and f 0 (t) exists and it is a
compact. The derivative f 0 (t) is a.a. if and only if, when [fa (t)]0 =
[f 0 (t)]a for every sequence a = an (in the case fa (t) exists).
Theorem 2. Let function f (t) be a.p. and f 0 (t) exists and it is continuous and compact. The derivative f 0 (t) is a.p. if and only if, when
[fa (t)]0 = [f 0 (t)]a for every sequence a = an (in the case fa (t) exists).
References
[1] Levitan B.M., Almost-Periodic Functions. Moscow, 1953. (in Russian)
[2] Veech W.A. Almost automorphic functions on groups. Amer. J. Math. 87(3)
(1965) 719–751.
[3] Dimitrova-Burlayenko S.D., The conditions for saving continuity for differentiating functions. Contemporary problems of mathematics, mechanics and
computing sciences. N.N. Kizilova, G.N. Zholtkevych (eds). Kharkov (2011)
332–338. (in Russian)
[4] Dimitrova-Burlaenko S.D., Almost automorphic functions as compact continuous functions on the group. Bulletin of National Technical University ’KhPI’.
Series: Mathematical modeling in engineering and technologies 27 (2012) 82–
85. (in Russian)
Date: July 30, 2015.
2000 Mathematics Subject Classification. 43A60.
1
2
S. D. DIMITROVA-BURLAYENKO
[5] Dimitrova-Burlaenko S.D., Necessary and sufficient conditions for convergence
of almost periodic functions to almost periodic functions. Contemporary problems of natural sciences 1(2) (2014) 100–104. (in Russian)
National Technical University Kharkiv Polytechnic Institute, Kharkiv,
Ukraine
E-mail address: s.dimitrova@mail.ru
A connected version of the Stone Duality Theorem
Georgi Dimov∗
“St. Kl. Ohridski” University of Sofia
Coauthor: Dimiter Vakarelov
The celebrated Stone Duality Theorem [St] states that the category Bool of all
Boolean algebras and Boolean homomorphisms is dually equivalent to the category
Stone of compact Hausdorff totally disconnected spaces and continuous maps. The
restriction of the Stone duality to the category CBool of complete Boolean algebras
and Boolean homomorphisms is a duality between the category CBool and the category of compact Hausdorff extremally disconnected spaces and continuous maps. We
introduce the notion of a Stone 2-space and the category 2-Stone of Stone 2-spaces
and suitable morphisms between them, and we show that the category 2-Stone is
dually equivalent to the category Bool. The Stone 2-spaces are pairs (X, X0 ) of a
compact connected T0 -space X and a dense subspace X0 of X, satisfying some mild
conditions. We introduce as well the notion of an extremally connected space and show
that the category ECS of extremally connected spaces and continuous maps between
them satisfying a natural condition, is dually equivalent to the category CBool. The
extremally connected space are compact connected T0 -spaces satisfying an additional
condition, and the open continuous maps are morphisms of the category ECS.
References
[St] M. H. Stone, The theory of representations for Boolean algebras. Trans. Amer.
Math. Soc., 40, 1936, 37–111.
∗
This talk was supported by the project no. 7/2015 “Contact algebras and extensions of topological
spaces” of the Sofia University “St. Kl. Ohridski”.
1
A UNIFIED APPROACH TO COMPACTNESS AND
COMPLETENESS
SZYMON DOLECKI
Completeness of a convergence is a notion relative to that of fundamental filter
with respect to a collection of (convergence) covers. If P is such a collection, then
a filter F is called P-fundamental if F ∩ P 6= ∅ for each P ∈ P; a convergence
is P-complete if each P-fundamental filter is adherent. The completeness number
compl (ξ) of a convergence ξ is the least cardinal such that there exists a collecton
of covers P of ξ of that cardinality, for which ξ is-complete. In this setting, a convergence ξ is countably complete if compl (ξ) ≤ ℵ0 (Čech-complete if ξ is a Tikhonov
topology), locally relatively compact whenever compl (ξ) < ℵ0 , thus compl (ξ) = 1,
compact if compl (ξ) = 0
Conditional completeness (and compactness) with respect to a class H of filters,
appears when the requirement of adherence is restricted to fundamental filters from
H; for instance, countably compact is conditionally compact with respect to the class
F1 of countably based filters, pseudocompleteness of Oxtoby [2] is F1 -conditional
countable completeness.
In this framework, P-completeness amounts to conditional compactness with
respect to P-fundamental filters.
This perspective enables one to unify the study of various variants of completeness and compactness, in particular, that of their preservation under operations on
convergences. For example, the Tikhonov theorem on compactness of products is
an immediate consequence of the following formula for completeness number [1]:
Y
X
compl
ξα =
compl (ξα ) .
α<κ
α<κ
References
[1] S. Dolecki. Elimination of covers in completeness. Topology Proceedings, 28:445–465, 2004.
[2] J. C. Oxtoby. Cartesian products of Baire spaces. Fund. Math., 49:157–166, 1960/1961.
Mathematical Institute of Burgundy, Université de Bourgogne Franche-Comté, Dijon, France
1
F -NODEC SPACES
LOBNA DRIDI, ABDELWAHEB MHEMDI, AND TAREK TURKI
Abstract. Following Van Douwen, a topological space is said to be nodec if
it satisfies one of the following equivalent conditions:
(i) every nowhere dense subset of X, is closed;
(ii) every nowhere dense subset of X, is closed discrete;
(iii) every subset containing a dense open subset is open.
This paper deals with a characterization of topological spaces X such that
F(X) is a nodec space for some covariant functor F from the category Top to
itself. T0 , ρ and FH functors are completely studied.
Secondly, we characterize maps f given by a flow (X, f ) in the category
Set such that (X, P(f )) is nodec (resp., T0 -nodec), where P(f ) is a topology
on X whose closed sets are precisely f -invariant sets.
(Lobna Dridi) Department of Mathematics, Tunis Preparatory Engineering Institute.
University of Tunis. 1089 Tunis, TUNISIA.
E-mail address, Lobna Dridi: lobna dridi 2006@yahoo.fr
(Abdelwaheb Mhemdi) Higher Institute of applied sciences and technologies of Gafsa,
bp 116, campus universitaire, 2112-Sidi Ahmed Zarroug Gafsa, TUNISIA.
E-mail address, Abdelwaheb Mhemdi: mhemdiabd@gmail.com
Department of Mathematics, Faculty of Sciences of Tunis. University Tunis-El Manar. “Campus Universitaire” 2092 Tunis, TUNISIA.
E-mail address, Tarek Turki: tarek turki@gmail.com
1
Topology in Art and Architecture
M. Emmer
Univ. Roma Sapienza & IVSLA, Venice, Italy
emmer@mat.uniroma1.it
“ The relation between the elements could be described as topological, in the
sense that it always seems possible to imagine a passage from one to the other by
stretching and twisting, shrinking and contracting ”, words by the artist Bruce
Nauman, who presented at the Venice Biennale of Art 2009 the exhibition
by the title Topological Gardens. The starting point to present relationships
between Topology, Art, Architecture and even Cinema in recent years.
1
Regular Homotopy Theory and the
Construction of the Derived Graphs
J. Carlos S. Kiihl∗
Abstract:
In this paper we present an overview of the Regular Homotopy Theory
for Digraphs, a survey of its main applications and the most recent developments. We show that the o-regular maps introduced by D. C. Demaria
coincides with the pre-continuos maps if, in a natural way, we introduce in
the class of the digraphs a structure of pre-topological space. We state and
present the basic concepts and the fundamental results of this Homotopy
Theory. New homotopical concepts and invariants associated to digraphs
are stablished and, using these new tools, a new approach to the study of
digraphs can be used, specially for the case of tournaments. For the Hamiltonian tournaments we have the concepts of: minimal cycles, cyclic characteristic, neutral and non-neutral vertices. One of the main purpose here is
to stablish the properties of these concepts which are important in order to
obtain structural characterizations for certain families of tournaments, when
they are approached from a homotopical point of view. As some important
applications we list some already known results about simply disconnected,
normal, Douglas tournaments and Moon tournaments. Then we give a complete study of the Hamiltonian 5-tournaments and 6-tournaments (having
the maximal number of non-neutral vertices) in terms of minimal 3-cycles.
This has led to the discovery of the construction of the Derived Graph associated to a Hamiltonian tournament. This most recent construction is offering
us a powerfull tool to continue our studies on the structural caracterization
of certain families of tournaments.
Keywords: Regular Homotopy Theory, Digraphs, Tournaments, Cycles,
Derived Graph.
MSC (2010) - 55Q99,05C20.
∗
jcarlos.kiihl@gmail.com
1
References
[1] BEINEKE, L. W. and REID, K. B., Tournaments–Selected Topics in
Graph Theory, Edited by L. W. Beineke and R. J. Wilson, Academic
Press, New York (1978), 169–204.
[2] BURZIO M. and DEMARIA D.C., A normalization theorem for regular
homotopy of finite directed graphs, Rend. Circ. Matem. Palermo, (2),
30 (1981), 255–286.
[3] BURZIO M. and DEMARIA D.C., The first normalization theorem for
regular homotopy of finite directed graphs, Rend. Ist. Mat. Univ. Trieste,
13 (1981), 38–50.
[4] BURZIO M. and DEMARIA D.C., The second and third normalization
theorem for regular homotopy of finite directed graphs, Rend. Ist. Mat.
Univ. Trieste, 15 (1983), 61–82.
[5] BURZIO, M. and DEMARIA, D. C., Duality theorem for regular homotopy of finite directed graphs, Rend. Circ. Mat. Palermo, (2), 31 (1982),
371–400.
[6] BURZIO, M. and DEMARIA, D. C., Homotopy of polyhedra and regular homotopy of finite directed graphs, Atti IIo Conv. Topologia, Suppl.
Rend. Circ. Mat. Palermo, (2), no¯ 12 (1986), 189–204.
[7] BURZIO, M. and DEMARIA, D. C., Characterization of tournaments
by coned 3–cycles, Acta Univ. Carol., Math. Phys., 28 (1987), 25–30.
[8] BURZIO, M. and DEMARIA, D. C., On simply disconnected tournaments, Proc. Catania Confer. Ars Combinatoria, 24 A (1988), 149–161.
[9] BURZIO, M. and DEMARIA, D. C., On a classification of hamiltonian
tournaments, Acta Univ. Carol., Math. Phys., 29 (1988), 3–14.
[10] BURZIO, M. and DEMARIA, D. C., Hamiltonian tournaments with the
least number of 3-cycles, J. Graph Theory 14 (6) (1990), 663–672.
[11] CAMION P., Quelques porpriété des chemins et circuits Hamiltoniens
dans la théorie des graphes, Cahiers Centre Études Rech. Oper., vol 2
(1960), 5–36.
2
[12] C̆ECH E.,Topological Spaces, Interscience, London (1966).
[13] DEMARIA D. C.; GARBACCIO BOGIN R., Homotopy and homology
in pretopological spaces, Proc. 11th Winter School, Suppl. Rend. Circ.
Mat. Palermo, (2), No¯ 3 (1984), 119-126.
[14] DEMARIA D.C.; GANDINI, P. M., Su una generalizzazione della teoria
dell’omotopia, Rend. Sem. Mat. Univ. Polit. Torino , 34 (1975 - 76).
[15] DEMARIA D.C.; GIANELLA G.M., On normal tournaments , Conf.
Semin. Matem. Univ. Bari ,vol. 232 (1989), 1-29.
[16] DEMARIA D.C.; GUIDO C., On the reconstruction of normal tournaments, J. Comb. Inf. and Sys. Sci., vol. 15 (1990), 301-323.
[17] DEMARIA D.C.; KIIHL J.C. S., On the complete digraphs which are
simply disconnected, Publicacions Mathemàtiques, vol. 35 (1991), 517525.
[18] DEMARIA D.C.; KIIHL J.C. S., On the simple quotients of tournaments that admit exactly one hamiltonian cycle, Atti Accad. Scienze
Torino, vol. 124 (1990), 94-108.
[19] DEMARIA D.C.; KIIHL J.C. S., Some remarks on the enumeration
of Douglas tournaments, Atti Accad. Scienze Torino , vol. 124 (1990),
169-185.
[20] DOUGLAS R.J., Tournaments that admit exactly one Hamiltonian circuit, Proc. London Math. Soc., 21, (1970), 716-730.
[21] GANDINI, P. M., Sull’omotopia per pseudoarchi, Rend. Sem. Mat.
Univ. Polit. Torino , 33 (1974 - 75).
[22] GIANELLA, G. M., Sull’omotopia per quasiarchi, Rend. Sem. Mat.
Univ. Polit. Torino , 31 (1971 - 72, 1972 - 73).
[23] GUIDO C., Structure and reconstruction of Moon tournaments, J.
Comb. Inf. and Sys. Sci., vol. 19 (1994), 47-61.
[24] GUIDO C., A larger class of reconstructable tournaments, Dicrete Math.
152 (1996), 171-184.
3
[25] GUIDO C.; KIIHL J.C.S.; OLIVEIRA, J. P. M.; BORRI, M., Some
remarks on non-reconstructable tournaments, (to appear).
[26] KIIHL, J. Carlos S.;GONÇALVES, A. C., On Digraphs and their Quotients, Revista Iluminart, Volume 9 (2012), 195 - 208.
[27] KIIHL, J. Carlos S.; GUADALUPE, Irwen Valle, Either Digraphs or
Pre-Topological Spaces?, Revista Iluminart, Volume 6 (2011), 129 - 147
[28] KIIHL, J. Carlos S.; TIRONI, Gino; GONÇALVES, A. C., The Minimal Cycles, Neutral and Non-Neutral Vertices in Tournaments, Revista
Iluminart, Volume 10 (2013), 213 - 238.
[29] KIIHL, J. Carlos S.; LIMA, F. M. B.; OLIVEIRA, J. P. M.;
GONÇALVES, A. C., 6-Tournaments having a minimal cycle of length
four, Revista Iluminart, Volume 12 (2013), 179 - 192.
[30] KIIHL, J. Carlos S.; GONÇALVES, Alexandre C., Hamiltonian Tournaments and Associated 3-Cycles Graphs, (to appear).
[31] KIIHL, J. Carlos S.;TIRONI, Gino, Non-Coned Cycles: A New Approach to Tournaments, Revista Iluminart, Volume 7 (2011), 98 - 109.
[32] MOON J.W., Topics on Tournaments, Holt, Rinehart and Winston,
New York (1978).
[33] MOON J.W., On subtournaments of a tournament, Canad. Math. Bull.,
vol. 9 (3) (1966), 297-301.
[34] MOON J.W., Tournaments whose subtournaments are irreducible or
transitive, Canad. Math. Bull., vol. 21 (1) (1979), 75-79.
[35] MÜLLER, V.; NES̆ETR̆IL J.; PELANT J., Either tournaments or algebras?, Discrete Math., 11 (1975), 37–66.
[36] STOCKMEYER P.K., The reconstruction conjecture for tournaments,
in “Proceedings, Sixth Southeastern Conference on Combinatorics,
Graph Theory and Computing” (F.Hoffman et al., Eds.), 561-566, Utilitas Mathematica, Winnipeg, (1975).
[37] STOCKMEYER P.K., The falsity of the reconstruction conjecture for
tournaments, J. Graph Theory 1 (1977), 19-25.
4
On function spaces for multifunctions
Ankit Gupta1 and Ratna Dev Sarma2
1
2
Department of Mathematics, University of Delhi, Delhi 110007, India
Department of Mathematics, Rajdhani College, University of Delhi, Delhi 110015, India.
email: 1 ankitsince1988@yahoo.co.in
2
ratna sarma@yahoo.com
Abstract
The interplay of the properties of a topological space X and those
of the function space C(X, Y ) of continuous functions from X to another space Y has been an area of active research in topology. Several
conditions under which the compact-open, Isbell or natural topologies
on the set of continuous real valued functions on a space may coincide have been studied in [4]. A unified theory of function spaces and
hyperspaces has been developed in [2]. In [3], it is shown that intersection of all admissible topologies on C(X, Y ) is admissible under certain
conditions. All these research papers are testimony of the keen interest
of the researchers in the study of function spaces.
In [4], a natural topology on the set of upper semi-continuous setvalued functions has been constructed. Apart from this, there is hardly
any discussion of the continuous multi-valued functions in the study of
function spaces. But the multi-valued functions are being rigorously
studied now a days in several other area of mathematics such as in
Optimization, Frame theory, Approximation theory etc.
In this paper, we bridge this gap by properly investigating the topological aspects of the function spaces for multifunctions. Starting from
the basic level, we define several topologies for continuous multifunctions. Point-open topology, open-open topology, compact-open topology etc are introduced and investigated. Unlike in [1], we have adopted
net theoretic approach to discuss continuous convergence for the topology of multifunctions. The net theory for sets are further developed for
this purpose. Conditions for splitting (resp. upper and lower splitting)
and admissibility (resp. upper and lower admissible) are obtained by
using the concept of continuous convergence. The characterizations
of admissibility and splitting using net theory as shown in Arens and
Dugundji [1] do not hold for multifunctions. Their variants are investigated in our paper. The compact-open topology for multifunctions
1
is upper splitting. While the point-open topology is found to be the
coarsest topology which is coordinate wise admissible, it is also the
finest topology which is coordinate wise splitting. Several examples
are provided to explain the intrinsic differences between the topologies
of continuous functions and topologies of continuous multifunctions.
References
[1] R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math.
1 (1951) 5-31.
[2] S. Dolecki, F. Mynard, A unified unified theory of function spaces and
hyperspaces: local properties, Houston journal of Math. 40(1) (2014),
285–318.
[3] D.N. Georgiou, S.D. Iliadis, On the greatest splitting topology, Topology
Appl. 156 (2008) 70-75.
[4] F. Jordan, Coincidence of function space topologies, Topology Appl. 157
(2010) 336–351.
Productivity of coreflective classes of
paratopological groups
M.Hušek
Department of Mathematics, Charles University, Prague
email: Miroslav.Husek@mff.cuni.cz
Abstract
Coreflective classes C in K (e.g., sequential spaces in K=Top) are
usually not productive in K (i.e., some products in K of κ-many spaces
from C do not belong to C). If such a least cardinal κ (called productivity number) is uncountable (i.e., C is countably productive) then
the coreflective class is productive in some models of set theory for
some K. Submeasurable and measurable cardinals play a role in such
situations.
At first we give a survey on productivity of coreflective classes in
topological and uniform spaces, in topological linear and locally convex
spaces, in topological groups. Then we show a situation for paratopological groups. For instance:
1. Productivity numbers of bicoreflective and non-productive subcategories of paratopological groups are submeasurable cardinals.
2. Every sequentially continuous homomorphism on a product of
paratopological groups into any paratopological group is continuous provided the coordinate spaces are sequential and the cardinality of the
index set of the product is non-sequential.
References
[1] B.Batı́ková, M.Hušek. Productivity numbers in paratopological groups.
Top.Appl., 193, 167–174, 2015.
1
On embeddings of topological groups of weight τ
S.D. Iliadis
Moscow State University (M.V. Lomonosov)
s.d.iliadis@gmail.com
Let S be a class of topological groups. It is said that a topological group K
is universal in this class if (a) K ∈ S and (b) for every X ∈ S, there exists a
topological subgroup of K which is topologically isomorphic to X.
V.V. Uspenskij (see [4], [5]) proved that in the class of all separable metrizable
topological groups there are universal elements. S.A. Shkarin see [3]) proved
that in the class of all separable metrizable topological Abelian groups there
exists a universal element. Moreover, he proved that, under GCH, for every
uncountable cardinal τ in the class of all metrizable topological Abelian groups
of weight ≤ τ and in the class of all topological Abelian groups of weight ≤ τ
there are universal elements. However, the problems of the existence of universal
elements in the class of all topological groups (see Question 2 of [5]) and in
the class of all metrizable topological groups (see Problem 4 of [3]) of a given
uncountable weight remain open.
Using the method of construction of so-called Containing Spaces given in [1], a
space of a given weight τ containing continuously the homeomorphic images
of all topological groups of weight at most τ , is constructed see [2].
Also, for given cardinals τ and µ, τ ≤ µ, we construct a topological group
of character τ and of weight 2µ containing topologically all topological groups
of character τ and of weight µ. In particular, If τ = ω, then there exists a
metrizable group of weight ≤ 2µ containing topologically all metrizable groups
of weight ≤ µ. From the using construction it follows automatically that the
above results are hold (without GCH) if all considered groups are Abelian.
References
[1] S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, 198. Elsevier Science B.V., Amsterdam, 2005. xvi+559 pp.
[2] Stavros Iliadis, On embeddings of topological groups, In print in the
journal Fundamental and Applied Mathematics.
[3] S.A. Shkarin, On universal Abelian topological groups, Matematicheskii
Sbornik, Vol. 190, No. 7 (1999),127-144.
[4] V.V. Uspenskij, A universal topological group with a countable base,
Funktsionanal’nyj analiz i ego prilozhenija (Functional analysis and its
applications) 20 (1986), 86-87.
[5] V.V. Uspenskij, On the group of isometries of the Urysohn universal
metric space, Comment.Math.Univ.Carolinae 31, 1(1990), 181-182.
1
On a new lower-Vietoris-type topology in hyperspaces
Elza Ivanova-Dimova∗
“St. Kl. Ohridski” University of Sofia
In 1975, M. M. Choban [Ch] introduced a new topology on the set of all closed
subsets of a topological space for obtaining a generalization of the famous Kolmogoroff
Theorem on operations on sets. This new topology is similar to the upper Vietoris
topology but is weaker than it. In 1998, G. Dimov and D. Vakarelov [DV] used a
generalized version of this new topology for proving an isomorphism theorem for the
category of all Tarski consequence systems. In this talk we will introduce a new
lower-Vietoris-type topology in a way similar to that with which the new upperVietoris-type topology was introduced in [DV]. We will study this new topology and,
in particular, we will generalize some results of [CMP].
References
[Ch] M. M. Coban, Operations over sets, Sibirsk. Mat. Z. 16 (1975), no. 6, 1332-1351.
[CMP] E. Cuchillo-Ibanez, M. A. Moron and F. R. Ruiz del Portal, Lower semifinite
topology in hyperspaces, Topology Proc. 17 (1992), 29-39.
[DV] G. Dimov and D. Vakarelov, On Scott consequence systems, Fundamenta Informaticae 33 (1998), 43-70.
∗
This talk was supported by the project no. 7/2015 “Contact algebras and extensions of topological
spaces” of the Sofia University “St. Kl. Ohridski”.
1
Regular Homotopy Theory and the
Construction of the Derived Graphs
J. Carlos S. Kiihl∗
Abstract:
In this paper we present an overview of the Regular Homotopy Theory
for Digraphs, a survey of its main applications and the most recent developments. We show that the o-regular maps introduced by D. C. Demaria
coincides with the pre-continuos maps if, in a natural way, we introduce in
the class of the digraphs a structure of pre-topological space. We state and
present the basic concepts and the fundamental results of this Homotopy
Theory. New homotopical concepts and invariants associated to digraphs
are stablished and, using these new tools, a new approach to the study of
digraphs can be used, specially for the case of tournaments. For the Hamiltonian tournaments we have the concepts of: minimal cycles, cyclic characteristic, neutral and non-neutral vertices. One of the main purpose here is
to stablish the properties of these concepts which are important in order to
obtain structural characterizations for certain families of tournaments, when
they are approached from a homotopical point of view. As some important
applications we list some already known results about simply disconnected,
normal, Douglas tournaments and Moon tournaments. Then we give a complete study of the Hamiltonian 5-tournaments and 6-tournaments (having
the maximal number of non-neutral vertices) in terms of minimal 3-cycles.
This has led to the discovery of the construction of the Derived Graph associated to a Hamiltonian tournament. This most recent construction is offering
us a powerfull tool to continue our studies on the structural caracterization
of certain families of tournaments.
Keywords: Regular Homotopy Theory, Digraphs, Tournaments, Cycles,
Derived Graph.
MSC (2010) - 55Q99,05C20.
∗
jcarlos.kiihl@gmail.com
1
References
[1] BEINEKE, L. W. and REID, K. B., Tournaments–Selected Topics in
Graph Theory, Edited by L. W. Beineke and R. J. Wilson, Academic
Press, New York (1978), 169–204.
[2] BURZIO M. and DEMARIA D.C., A normalization theorem for regular
homotopy of finite directed graphs, Rend. Circ. Matem. Palermo, (2),
30 (1981), 255–286.
[3] BURZIO M. and DEMARIA D.C., The first normalization theorem for
regular homotopy of finite directed graphs, Rend. Ist. Mat. Univ. Trieste,
13 (1981), 38–50.
[4] BURZIO M. and DEMARIA D.C., The second and third normalization
theorem for regular homotopy of finite directed graphs, Rend. Ist. Mat.
Univ. Trieste, 15 (1983), 61–82.
[5] BURZIO, M. and DEMARIA, D. C., Duality theorem for regular homotopy of finite directed graphs, Rend. Circ. Mat. Palermo, (2), 31 (1982),
371–400.
[6] BURZIO, M. and DEMARIA, D. C., Homotopy of polyhedra and regular homotopy of finite directed graphs, Atti IIo Conv. Topologia, Suppl.
Rend. Circ. Mat. Palermo, (2), no¯ 12 (1986), 189–204.
[7] BURZIO, M. and DEMARIA, D. C., Characterization of tournaments
by coned 3–cycles, Acta Univ. Carol., Math. Phys., 28 (1987), 25–30.
[8] BURZIO, M. and DEMARIA, D. C., On simply disconnected tournaments, Proc. Catania Confer. Ars Combinatoria, 24 A (1988), 149–161.
[9] BURZIO, M. and DEMARIA, D. C., On a classification of hamiltonian
tournaments, Acta Univ. Carol., Math. Phys., 29 (1988), 3–14.
[10] BURZIO, M. and DEMARIA, D. C., Hamiltonian tournaments with the
least number of 3-cycles, J. Graph Theory 14 (6) (1990), 663–672.
[11] CAMION P., Quelques porpriété des chemins et circuits Hamiltoniens
dans la théorie des graphes, Cahiers Centre Études Rech. Oper., vol 2
(1960), 5–36.
2
[12] C̆ECH E.,Topological Spaces, Interscience, London (1966).
[13] DEMARIA D. C.; GARBACCIO BOGIN R., Homotopy and homology
in pretopological spaces, Proc. 11th Winter School, Suppl. Rend. Circ.
Mat. Palermo, (2), No¯ 3 (1984), 119-126.
[14] DEMARIA D.C.; GANDINI, P. M., Su una generalizzazione della teoria
dell’omotopia, Rend. Sem. Mat. Univ. Polit. Torino , 34 (1975 - 76).
[15] DEMARIA D.C.; GIANELLA G.M., On normal tournaments , Conf.
Semin. Matem. Univ. Bari ,vol. 232 (1989), 1-29.
[16] DEMARIA D.C.; GUIDO C., On the reconstruction of normal tournaments, J. Comb. Inf. and Sys. Sci., vol. 15 (1990), 301-323.
[17] DEMARIA D.C.; KIIHL J.C. S., On the complete digraphs which are
simply disconnected, Publicacions Mathemàtiques, vol. 35 (1991), 517525.
[18] DEMARIA D.C.; KIIHL J.C. S., On the simple quotients of tournaments that admit exactly one hamiltonian cycle, Atti Accad. Scienze
Torino, vol. 124 (1990), 94-108.
[19] DEMARIA D.C.; KIIHL J.C. S., Some remarks on the enumeration
of Douglas tournaments, Atti Accad. Scienze Torino , vol. 124 (1990),
169-185.
[20] DOUGLAS R.J., Tournaments that admit exactly one Hamiltonian circuit, Proc. London Math. Soc., 21, (1970), 716-730.
[21] GANDINI, P. M., Sull’omotopia per pseudoarchi, Rend. Sem. Mat.
Univ. Polit. Torino , 33 (1974 - 75).
[22] GIANELLA, G. M., Sull’omotopia per quasiarchi, Rend. Sem. Mat.
Univ. Polit. Torino , 31 (1971 - 72, 1972 - 73).
[23] GUIDO C., Structure and reconstruction of Moon tournaments, J.
Comb. Inf. and Sys. Sci., vol. 19 (1994), 47-61.
[24] GUIDO C., A larger class of reconstructable tournaments, Dicrete Math.
152 (1996), 171-184.
3
[25] GUIDO C.; KIIHL J.C.S.; OLIVEIRA, J. P. M.; BORRI, M., Some
remarks on non-reconstructable tournaments, (to appear).
[26] KIIHL, J. Carlos S.;GONÇALVES, A. C., On Digraphs and their Quotients, Revista Iluminart, Volume 9 (2012), 195 - 208.
[27] KIIHL, J. Carlos S.; GUADALUPE, Irwen Valle, Either Digraphs or
Pre-Topological Spaces?, Revista Iluminart, Volume 6 (2011), 129 - 147
[28] KIIHL, J. Carlos S.; TIRONI, Gino; GONÇALVES, A. C., The Minimal Cycles, Neutral and Non-Neutral Vertices in Tournaments, Revista
Iluminart, Volume 10 (2013), 213 - 238.
[29] KIIHL, J. Carlos S.; LIMA, F. M. B.; OLIVEIRA, J. P. M.;
GONÇALVES, A. C., 6-Tournaments having a minimal cycle of length
four, Revista Iluminart, Volume 12 (2013), 179 - 192.
[30] KIIHL, J. Carlos S.; GONÇALVES, Alexandre C., Hamiltonian Tournaments and Associated 3-Cycles Graphs, (to appear).
[31] KIIHL, J. Carlos S.;TIRONI, Gino, Non-Coned Cycles: A New Approach to Tournaments, Revista Iluminart, Volume 7 (2011), 98 - 109.
[32] MOON J.W., Topics on Tournaments, Holt, Rinehart and Winston,
New York (1978).
[33] MOON J.W., On subtournaments of a tournament, Canad. Math. Bull.,
vol. 9 (3) (1966), 297-301.
[34] MOON J.W., Tournaments whose subtournaments are irreducible or
transitive, Canad. Math. Bull., vol. 21 (1) (1979), 75-79.
[35] MÜLLER, V.; NES̆ETR̆IL J.; PELANT J., Either tournaments or algebras?, Discrete Math., 11 (1975), 37–66.
[36] STOCKMEYER P.K., The reconstruction conjecture for tournaments,
in “Proceedings, Sixth Southeastern Conference on Combinatorics,
Graph Theory and Computing” (F.Hoffman et al., Eds.), 561-566, Utilitas Mathematica, Winnipeg, (1975).
[37] STOCKMEYER P.K., The falsity of the reconstruction conjecture for
tournaments, J. Graph Theory 1 (1977), 19-25.
4
Computing of the coarse shape groups
N. Koceic Bilan
Department of Mathematics, University of Split, Croatia
email: koceic@pmfst.hr.
Abstract
The coarse shape groups are new topological invariants which are
(coarse) shape and homotopy invariants, as well. Their structure is
signi…cantly richer than the structure of shape groups. They provide
information (especially, about compacta) even better than the homotopy pro-groups. Since nontrivial coarse shape groups, even for polyhedra, are too large, it is di¢ cult to calculate them exactly. In this
talk, we give an explicit formula for computing coarse shape groups of a
large class of metric compacta including solenoids. Moreover, we show
that every coarse shape group can be obtained as the inverse limit of
an inverse system of groups. It is proven that, for inverse systems of
compact polyhedra, the coarse shape group functor commutes with the
inverse limit. These results provide computing of coarse shape groups
in an easier manner.
References
[1] N. Koceic Bilan. Computing coarse shape groups of solenoids. Math.
Commun. 14 (2014), 243-251.
[2] N. Koceic Bilan. The coarse shape groups. Topology and its Applications.
157 (2010) 894–901.
1
On off-diagonal Fσ -δ-normality and on δ-normality of hyperspace
Anatoly Kombarov
Moscow State University, Moscow 119991, Russia
kombarov@mech.math.msu.su
Let P be a topological property. A space X has the off-diagonal property
P if X 2 \ ∆ has P. A space is said to be δ-normal if any two disjoint closed
sets, of which one is a regular Gδ , can be separated by disjoint open sets [1].
A subset G of a topological space is a regular Gδ if it is the intersection of
the closures of a countable collection of open sets each of which contains G. A
space X is called Fσ -δ-normal if all Fσ -sets in X are δ-normal [2]. It is known
that every off-diagonal normal compact space is first-countable [3] and more
generally every off-diagonal normal countably compact space is first-countable
[4].
Theorem 1. Every off-diagonal Fσ -δ-normal regular countably compact space
X is first-countable at every point x in a dense subset of X.
In particular, every off-diagonal Fσ -δ-normal dyadic space is metrizable.
It is known that if exp(X) is Fσ -δ-normal, then X is a compact space [2].
Here the space exp(X) is the set of all nonempty closed subsets of X with the
Vietoris topology. The space exp(ω1 ) is countably compact and hence δ-normal,
but ω1 is not compact.
Theorem 2. The space exp(ω) is not δ-normal.
Theorem 2 is a simultaneous generalization of the next two theorems: the
space exp(ω) is not normal [5] and is not countably paracompact [6].
References
[1]
[2]
[3]
[4]
[5]
[6]
J. Mack, Trans. Amer. Math. Soc. 148 (1970) 265–272.
A. P. Kombarov, Topology Appl. 91 (1999) 11–15.
A. V.Arhangel’skii , A. P.Kombarov, Topology Appl. 35 (1990) 121–126.
D. V. Malyhin, Vestn. Mosk. Univ., Matem. Mehan. 1997, N 5, 31–33.
V. M. Ivanova, DAN SSSR 101 (1955) 601–603.
N. Kemoto, Topology Appl. 154 (2007) 358–363.
1
Wilder continua and their subfamilies as absorbers
K. Królicki and Pawel Krupski
Mathematical Institute, University of Wroclaw
email: pawel.krupski@math.uni.wroc.pl
Abstract
A nondegenerate continuum X is said to be a Wilder continuum
if for any distinct points x, y, z ∈ X there is a subcontinuum of X
containing x and exactly one of the points y, z. We consider Wilder
continua in the cube I n , 3 ≤ n ≤ ∞, and their subfamilies: continuumwise Wilder continua, hereditarily arcwise connected continua, aposyndetic or colocally connected continua. The first three collections are
coanalytic absorbers in the hyperspace C(I n ) of subcontinua of I n ,
whereas two last ones are Fσδ -absorbers.
1
Sami Lazaar, Facult of sciences of Gafsa, Tunisia
Abstract
In the paper published in Topology Proceeding (2008), Lazaar and
Echi show that the orthogonal of the category of Tychonoff spaces is
exactly the family of all continuous maps orthogonal to the real line
R which are ρ-bijective.An interessant question cited in this paper
is the following: It is possible to remove the condition ρ-bijective in
order to obtain the orthogonality of the category TYCH is exactly the
real line orthogonal and consequently an interesting characterization
of Tychonoff spaces using orthogonality. A complete answer of this
question is given.
1
A metrizable X with Cp(X) not homeomorphic to
Cp(X) × Cp(X)
Witold Marciszewski
University of Warsaw
email: wmarcisz@mimuw.edu.pl
Abstract
We give an example of an infinite metrizable space X such that
the space Cp (X) of continuous real-valued functions on X endowed
with the pointwise topology, is not homeomorphic to its own square
Cp (X)×Cp (X). The space X is a zero-dimensional subspace of the real
line. Our result answers a long-standing open question in the theory
of function spaces posed by A.V. Arhangel’skii.
This is a joint research with Mikolaj Krupski.
1
On inverse limit of self-covering spaces
Vlasta Matijević
Faculty of science, University of Split
email: vlasta@pmfst.hr
A compact connected Hausdor¤ space Y is called self-covering provided
that whenever Y admits a …nite-sheeted covering map f : X ! Y from
a connected space X; then X is homeomorphic to Y: Note that; for each
n 2 N; the n-torus T n (i.e: the product of n copies of the unit circle S 1 )
is an abelian topological group and a self-covering metric continuum. We
consider the following question:
Let Y be the inverse limit of an inverse sequence; where each term is a
self-covering metric continuum and each bonding map is an open surjection.
Is Y a self-covering space ?
In order to answer the question we study …nite-sheeted covering maps
over compact connected n-dimensional abelian groups Y; where the groups
Y are obtained as the inverse limits of inverse sequences where each term is
T n and each bonding map is a …nite-sheeted covering homomorphism over
T n : Such groups Y are called solenoids for n = 1 and toroids for n = 2:
First we show that any solenoid is a self-covering space. However; for each
prime p > 2 and a non-quadratic p-adic integer ; we construct a toroid
Y ( ) which admits a 4-sheeted covering map f : X ! Y ( ); where X
is connected and non-homeomorphic to Y ( ). In this way we answer the
question in the negative.
This is a joint work with Katsuya Eda.
1
Nonhomogeneity of remainders
Jan van Mill
University of Amsterdam
Coauthor: A. V. Arhangel’skii
A space X is homogeneous if for any two points x, y ∈ X there is a homeomorphism h
from X onto itself such that h(x) = y.
In 1956, Walter Rudin proved that the Čech-Stone remainder βN \ N, where N is the
discrete space of positive integers, is not homogeneous under CH. This result was later
generalized considerably by Frolı́k who showed in ZFC that βX \ X is not homogeneous,
for any nonspeudocompact space X. Van Douwen and Kunen proved many results that
are in the same spirit.
Hence the study of (non)homogeneity of Čech-Stone remainders has a long history. In
this talk we are interested in homogeneity properties of arbitrary remainders of topological
spaces. We address the following general problem: when does a space have a homogeneous
remainder? If X is locally compact, then the Alexandroff 1-point compactification αX of
X has a homogeneous remainder. Hence for locally compact spaces, our question has an
obvious answer. If X is not locally compact, however, then it need not have a homogeneous remainder, as the topological sum of the space of rational numbers and the space
of irrational numbers shows. Hence we consider questions of the following type: if X is
homogeneous, and not locally compact, does X have a homogeneous remainder? We will
present several cardinal inequalities for the number of homeomorphisms of a remainder of
a nowhere locally compact space. That our bounds are independent, is demonstrated by
examples. As an application, it follows that if X is countable and nowhere locally compact,
then any remainder of X has at most c homeomorphisms, where c denotes the cardinality
of the continuum. From this we get an example of a countable topological group G no
remainder of which is homogeneous. We also present an example of a separable metrizable
topological group, no remainder of which is homogeneous.
1
Topoloical properties of orbits and orbit spaces of
some foliations on manifolds
R. Mirzaei
Department of Mathematics, I. Kh. International University, Qazvin, Iran
email: r.mirzaei@sci.ikiu.ac.ir, r− mirzaioe@yahoo.com
Abstract
Let G × M → M be a differentiable action of a Lie group G on a
differentiable manifold M and consider the orbit space M
G with the quois
called
the
cohomogeneity
of the
tient topology. The dimension of M
G
action of G on M . Study of orbit spaces has many important applications in invariant function theory and G-invariant variational problems
associated to M . Many G-invariant objects associated to M can be
related to similar objects associated to the orbit space. Therefore, if
the dimension of the orbit space is small enough, we can effectively
reduce many problems about G-invariant objects of M to generally
easier problems on M
G . Because of this motivation, many mathematicians studied topological properties of the orbit spaces of Lie group
actions on manifolds. In special case, when M is a Riemannian manifold and G is a connected and closed subset of the isometries of M ,
the action induces a singular foliation on M . If dim M
G = 1 then one
can find a lot of papers about the orbits and orbit spaces. But, there
are many important open problems to solve in the case dim M
G > 1.
We give some results about topological properties of orbits and orbit spaces of cohomogeneity two actions on Riemannian manifolds of
nonpositive curvature.
References
[1] Bredon. G. E, Introdution to compact transforation groups, Acad.
Press, New york , London, 1972.
[2] Brendt J. ; Console S. ; Olmos C., Submanifolds and holonomy, Chapman and Hall/CRS. London, New yourk, 2003.
[3] Kobayashi S. ; Nomizu K., Foundations of differential geometry, Vol. I,
II, Wiely Interscience, New York, 1963, 1969.
1
[4] Michor P.W, Isomrtric actions of Lie groups and invariants ,
Lecture course at the university of Vienna, 1996/97, http :
//www.mat.univie.ac.at/ michor/tgbook.ps
[5] Mirzaie R., Cohomogeneity two actions on flat Riemannian manifolds,
Acta mathematica sinica (Engl. se.) 23(9)(2007) 1587-1592.
[6] Mirzaie R., On negatively curved Riemannian manifolds of low cohomogeneity, Hokkaido math. journal 38(2009) 797-803.
[7] Mirzaie R. ; Kashani S. M. B., On cohomogeneity one flat Riemannian
manifolds, Glasgow Math. J., 44(2002) 185-190.
[8] Mostert P., On a compact Lie group action on manifilds, Ann. Math.
65(1957) 447-455.
[9] Palais R. S. ; Terrg Ch. L., A genereal theory of canonical forms, Am.
Math. Soc., 300(1987) 771-789.
[10] Podesta F. ; Spiro A., Some topological properties of cohomogeneity one
manifolds with negative curvature, Ann. Global. Anal. Geom, 14(1966)
69-79.
[11] Searle C., Cohomogeneity and positive curvature in low dimensions,
Math. Z., 214(1993) 491-498.
[12] Wolf J. A., Spaces of constant curvature, Berkely, California, 1977 . . .
Subelliptic equations
on Carnot groups
G. Molica Bisci
Department PAU, University of Reggio Calabria
email: gmolica@unirc.it
Abstract
It is well-known that a great attention has been focused by many
authors on the study of subelliptic equations on Carnot groups and in
particular on the Heisenberg group Hn . See, among others, the papers
[1, 2, 3, 4] and references therein.
Motivated by this large interest in the literature, we are interested
on the existence of weak solutions for the following problem
{
−∆G u = λf (ξ, u) in D
(Pλf )
u|∂D = 0,
where D is a smooth bounded domain of the Carnot group G, ∆G is
the Kohn-Laplacian on G, and λ is a positive real parameter.
References
[1] M.Z. Balogh and A. Kristály, Lions-type compactness and Rubik
actions on the Heisenberg group, Calc. Var. Partial Differential Equations 48 (2013), 89-109.
[2] S. Bordoni, R. Filippucci, and P. Pucci, Nonlinear elliptic inequalities with gradient terms on the Heisenberg group, Nonlinear Analysis (2015), http://dx.doi.org/10.1016/j.na.2015.02.012
[3] N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ.
Math. J. 41 (1992), 71-98.
[4] A. Loiudice, Semilinear subelliptic problems with critical growth on
Carnot groups, Manuscripta Math. 124 (2007), 247-259.
1
Khovanov Homology of Braid Links
A. R. Nizami, M. Munir, Ammara Usman
Division of Science and Technology, University of Education, Township Campus, Lahore-Pakistan
email: arnizami@ue.edu.pk, mmunir@ue.edu.pk, ammarausman88@ymail.com
Abstract
Although computing the Khovanov homology of links is common
in literature, no general formulae have been given for all of them. We
give the general formulae of the Khovanov homology of the 2-strand
cn1 and the 3-strand braid link ∆2k , where ∆ = x1 x2 x1 .
braid link x
References
[1] Alexander J. Topological invariants of knots and links. Trans Amer
Math Soc, 20, 275-306, 1923.
[2] Artin E. Theory of braids. Annals of Mathematics,48, 101-126, 1947.
[3] Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebraic and Geometric Topology, 2, 337-370, 2002.
[4] V. Jones, A polynomial invariant for knots via Von Neumann algebras,
Bull. Amer. Math. Soc., vol.12, 1, 103111, 1985.
[5] Kauffman L. H. State Models and the Jones Polynomial. Topology, 26,
395-407, 1987.
[6] Mikhail Khovanov, A categorification of the Jones polynomial, Duke
Mathematical Journal 101 no.3, 359-426, 2000.
[7] Vassily Manturov, Knot Theory, Chapman and Hall/CRC, 2004.
[8] Reidemeister K. Knot Theory. New York, USA: Chelsea Publ and Co,
1948.
[9] K. Reidemeister: Elementary begrundung der knotentheorie, Abh.
Math. Sem. Univ. Hamburg 5, 24-32, 1926.
CHARACTERISATIONS OF BOUNDARY FRAMES
JISSY NSONDE-NSAYI
Abstract. We say a completely regular frame L is a boundary frame if for every cozero
element c of L there exists a dense cozero element d of L such that d ≤ c ∨ c∗ . This
generalise the notion of ∂-space defined by Azarpanah and Karavan in [1]. Our goal in
this talk is to give ring theoretic and localic characterisations of boundary frames.
References
[1] F. Azarpanah and M. Karavan, On nonregular ideals and z ◦ -ideals in C(X), Czech. Math. J. 55
(2005), 397–407.
1
Topology of Vitali selectors on the real line
Venuste Nyagahakwa and Vitalij A. Chatyrko
Department of Mathematics, Linköping University
venuste.nyagahakwa@liu.se and vitalij.tjatyrko@liu.se
Abstract
Let F be the family of all dense countable subgroups of the real numbers R. Consider Q ∈ F.
Recall that a Vitali Q-selector of R is any set which meets every coset x + Q, x ∈ R, precisely in one
point. We denote by V(Q) the family of all Vitali Q-selectors of R and by SV(Q) the family of all
non-empty finite unions of elements of V(Q). Let us recall (see [2] and [1], resp.) that the elements
of SV(Q) are non-measurable in the Lebesgue sense and without the Baire property on the real line.
We define a new topology τ (Q) on R generated by the family {R\S : S ∈ SV(Q) } as a base. One can
observe that a subset A of R is closed according to this topology iff sup{|A∩(x+Q) : x ∈ R|} < ℵ0 or
A = R. We study topological properties of spaces R(Q) = (R, τ (Q)), Q ∈ F. In particular, we show
that each R(Q) is a T1 (not T2 ) hyperconnected topological space with ind R(Q) = Ind R(Q) = 1.
Moreover, if Q1 , Q2 ∈ F then the spaces R(Q1 ) and R(Q2 ) are homeomorphic. Let us note that if
Q1 ⊆ Q2 and |Q2 /Q1 | = ℵ0 then SV(Q1 ) ∩ SV(Q2 ) = ∅, and there exists a countable subset A of R
such that A is closed in R(Q1 ) but A is neither closed nor open in R(Q2 ).
The results above can be extended to abelian Hausdorff topological groups of the second category without isolated points having countable dense subgroups.
References
[1] V. A. Chatyrko, On Vitali sets and their unions, Matematicki Vesnik, 63, 2 (2011) 87-92
[2] A. B. Kharazishvili, Measurability properties of Vitali sets, Amer. Math. Monthly 118 (2011),
no. 8, 693-703
1
Compactifications of ω and the Banach space c0
Grzegorz Plebanek
Instytut Matematyczny, Uniwersytet Wroclawski
email: gplebanek@gmail.com
Abstract
We address the problem if there is in ZFC a compactification γω
of ω such that its remainder γω \ ω is nonseparable and carries a
strictly positive measure. Such a compactification can be constructed
assuming a relatively mild set-theoretic assumption.
Given γω, the space of continuous functions C(γω) contains a natural copy X of c0 , where
X = {f ∈ C(γω) : f |γω \ ω = 0}.
We investigate for which γω the space X is complemented in C(γω).
This is the case if γω is metrizable (Sobczyk’s theorem); on the other
hand, X is not complemented in C(βω) by a theorem due to Phillips.
If the space X is complemented in C(γω) then there is a strictly
positive measure on γω \ ω. We show that under CH the reverse implication does not hold and present another related example. Joint
research with Piotr Drygier (Wroclaw).
1
Open tilings and open monotonic tilings of
topological spaces
G. M. Reed
St Edmund Hall, Oxford University (emeritus fellow)
email: georgemichael.reed@gmail.com
Abstract
The goal is to develop a concept of ”tiling” a given topological space
with tiles of richer structure.
A space has an open tiling (OT) if for each open set U, there exists
a collection T of mutually disjoint open sets such that the closure of
each member of T is contained in U, and U is equal to the union of
the closures of the members of T.
An α-open tiling (α-OT) of a space is a an OT in which each open
set has a tiling of cardinality less than or equal to α.
A space has an α-open monotonic tiling (α-OMT) of type K if for
each open set U there exists a increasing chain C w.r.t. set inclusion
of open subsets of U such that each member of C has the topological
property K and U is equal to the union of the members of C, where
the cardinality of C is less than or equal to α.
In this talk we restrict space to mean a regular, Hausdorff, first
countable space, and consider which spaces have an OT, omega-OT,
or omega-OMT of type K. We also consider which non-Moore spaces
can be tiled by Moore tiles, and which non-metrisable spaces can be
tiled by metrisable tiles.
A few basic theorems are given, and a wide variety of standard
examples are examined. The results are surprising, at least to the
author.
A major open question concerns the existence of a normal, nonmetrisable space which has an ω-OMT of metrisable type. The author
first raised this question in the 1990s, and it has often been raised in
survey articles. The author has shown that:
(1) Such a space can not be countably-metacompact.
(2) Such a space must be CWH.
(3) Such a space would be a Dowker space with a σ-disjoint base.
(4) Under MA or (b=c), such a space must have cardinality ≥ c.
1
(5) There exists a space with an ω-OMT of metrisable type which
is not a Moore space.
(6) Results (1), (3), (4), and (5) would hold for the existence of
a normal, non-metrisable space which is the union of countably may
open metrisable sets. The author has referred to this question as the
normal, non-moore space problem.
In classical Moore tradition, the author suggests that readers should
find their own solutions. It is fun.
Comparability of bornological convergences on
the hyperspace of a uniformizable space
Marco Rosa∗
Paolo Vitolo∗
marco.rosa@unibas.it
paolo.vitolo@unibas.it
Abstract
Given a compatible uniformity U and an arbitrary bornology S on a topological
space X, we can define on the hyperspace CL(X) of non-empty closed subsets
of X, three convergences related to U and S: the upper, lower and “two-sided”
bornological convergence ([1, 2, 3]). Bornological convergence is a generalization
of the well known Attouch–Wets convergence.
We consider two compatible uniformities U and V, two arbitrary bornologies
S and T on X and we give necessary and sufficient conditions for the comparability of the lower, upper and “two-sided” bornological convergences they
generate.
We also focus on the particular case of the bornology of “Bourbaki bounded”
sets with respect to a uniformity.
Moreover, we consider the bounded-proximal topology generated by an arbitrary bornology and by the proximity induced by a compatible uniformity. We
characterize the comparability of convergences induced by bounded-proximal
topologies related to two compatible uniformities and two arbitrary bornologies.
Keywords: Hyperspace, Attouch-Wets Topology, Bornological Convergence,
Uniform Space, Bounded-proximal Topology.
MSC[2010]: Primary 54B20, Secondary 54E15, 54A20.
References
[1] Gerald Beer, Camillo Costantini, and Sandro Levi. Bornological convergence
and shields. Mediterr. J. Math., 10(1):529–560, 2013.
[2] Gerald Beer and Sandro Levi. Pseudometrizable bornological convergence
is Attouch-Wets convergence. J. Convex Anal., 15(2):439–453, 2008.
[3] A. Lechicki, S. Levi, and A. Spakowski. Bornological convergences. J. Math.
Anal. Appl., 297(2):751–770, 2004. Special issue dedicated to John Horváth.
∗ Dipartimento di Matematica, Informatica ed Economia, Università degli studi della Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza (Italy)
1
The Menger property, function spaces and the sequential fan
Masami Sakai
Kanagawa University, Japan
sakaim01@kanagawa-u.ac.jp
We discuss why the Menger property of Cp (X) implies finiteness of X, and
when the sequential fan Sω can be embedded into Cp (X).
1
Complete solution of Markov’s problem on the existence of
connected Hausdorff group topologies
Dmitri Shakhmatov
Ehime University, Japan
E-mail: dmitri.shakhmatov@ehime-u.ac.jp
This is a joint work with Dikran Dikranjan (Udine University, Italy).
It is easy to see that a non-trivial connected Hausdorff group must have cardinality at least continuum. Seventy years ago Markov asked if every group of
cardinality at least continuum can be equipped with a connected Hausdorff group
topology. Twenty five years ago a counter-example to Markov’s conjecture was
found by Pestov, and a bit later Remus showed that no permutation group admits
a connected Hausdorff group topology. The question (explicitly asked by Remus)
whether the answer to Markov’s question is positive for abelian groups remained
widely open. We prove that every abelian group of cardinality at least continuum
has a connected Hausdorff group topology, Furthermore, we give a complete characterization of abelian groups which admit a connected Hausdorff group topology
having compact completion.
1
The weak Whyburn spaces and cardinality.
Santi Spadaro
IME - Instituto de Matematica e Estatistica - University of Sao Paulo, Brazil
santispadaro@yahoo.com
The weak Whyburn property is a convergence property that has received a
lot of attention in the last twenty years. We compare it with the other convergence properties, in particular with the pseudoradial property. For example,
we show that every regular weakly Whyburn P-space of countable extent and
character at most ℵ2 is pseudo radial and that every Urysohn countably compact space of cardinality smaller than the continuum is weakly Whyburn. We
also construct examples of a countably compact Hausdorff non-weakly Whyburn
space of cardinality ω1 (showing that the Urysohn property is essential in the
latter result when the continuum hypothesis fails) and of a countably compact
regular weakly Whyburn non-pseudoradial space (which answers to a question
asked by Angelo Bella in private communication).
This is a joint work with D. A. Lara.
1
Separation and Cardinality
M. Bonanzinga, D. Stavrova, P. Staynova
Department of Mathematics, University of Leicester
email: ds311@le.ac.uk
Abstract
This is joint work with M. Bonanzinga and P. Staynova.
We continue studying combinatorial separation axioms as defined in [Bon13], [BSS15] and
their reflection on cardinal invariant inequalities. All notations are as in [Juh80], [Hod84].
We define a new cardinal function related to the pseudocharacter and Hausdorff properties as
follows: let X be a T1 topological space and for all x ∈ X, let
\
Hw(x) = {U : U ∈ Ux , |Ux | 6 ψ(X), Ux is a neighbourhood system of x}.
The Hausdorff width is HW (X) = sup{|Hw(x)| : x ∈ X}. We use it to generalise several results
about cardinality of not necessarily Hausdorff spaces. Amongst them is:
1 Theorem. If X is a T1 n-Hausdorff space, then |X| 6 HW (X)2aLc (X)χ(X) .
This result transfers to non-Hausdorff spaces the main result in [WD84].
The next result generalises the famous theorem of de Groot [dG65]:
2 Theorem. If X is T1 then |X| 6 HW (X)ψ(X)haL(X) .
We also improve for n-Hausdorff spaces a recent result of [JSS]:
pd(X)
3 Theorem. Let X be an n-Hausdorff space. Then |X| 6 22
.
Here, the pinning number pd(X) is defined in [JSS] and is less than the density.
Productivity of n-Hausdorff number has also been studied.
Many examples and open questions related to the structure of the spaces satisfying combinatorial separation axioms have been considered.
References
[Bon13] M. Bonanzinga. On the Hausdorff number of a topological space. Houston Journal of
Mathematics, 39(3):1013–1030, 2013.
[BSS15] M. Bonanzinga, D. Stavrova, and P. Staynova. Combinatorial separation axioms. Topology
and Its Applications, 2015. to appear.
[dG65] J. de Groot. Discrete subspaces of Hausdorff spaces. Bull. Acad. Polon. Sci., 13:537–544,
1965.
[Hod84] R. E. Hodel. Cardinal functions I. In K. Kunen and J. E. Vaughan, editors, Handbook of
Set Theoretic Topology, pages 1–61. Elsevier, North Holland, Amsterdam, 1984.
1
[JSS]
I. Juhasz, L. Soukup, and Z. Szentmiklossy. Pinning down versus density. to appear.
[Juh80] I. Juhasz. Cardinal Functions in Topology - ten years later. Mathematical Centre Tracts
123, Amsterdam, 1980.
[WD84] S. Willard and U. N. B. Dissanayake. The almost Lindelöf degree. Canad. Math. Bull.,
27(4), 1984.
2
A NEW PRESENTATION OF THE CATEGORY Ho(PRO(TOP))
L.STRAMACCIA - Università di Perugia
Inverse systems have been widely used in Mathematics, especially in Topology.
Grothendieck
was the first to give a good categorical definition for the category
PRO(C) of inverse systems in a given category C. The need for a homotopy theory of
PRO(C) was early recognized. Many authors were then concerned with the task of
defining a Quillen model structure on PRO(C) assuming C had one, in order to obtain
a well behaved homotopy category. The so called Steenrod homotopy category
Ho(PRO(TOP)) was defined by Porter and Edwards-Hastings about 1976. In the last
years further work on the subject has been done notably by Isaksen, and very
recently by Descotte and Dubuc .
There are at hand essentially two ways to look at Ho(PRO(TOP)). The first one is
that of Porter and Edwards-Hastings who define it by localizing PRO(TOP) at the
class of level equivalences so that in this case the morphisms are quite ugly to
handle. The second one is due to Cathey- Segal : given inverse systems X, Y in
TOP, they consider suitable fibrant replacements X’, Y’ for them obtaining that
Ho(PRO(TOP))(X,Y)= [X’, Y’ ],
where the right member denotes the set of homotopy classes. In this case
morphisms are easy to manage while the constructions of the fibrant replacements
is not trivial at all. We construct a new category cPRO(TOP) with objects the
inverse systems in TOP having the advantages of both the points of view above. We
prove that cPRO(TOP) is isomorphic to the Steenrod homotopy category
Ho(PRO(TOP)) and also to the coherent pro-homotopy category CHPRO(TOP) as
defined by Lisica-Mardesic.
Products of Menger sets
Piotr Szewczak∗
Cardinal Stefan Wyszyński University in Warsaw
email: p.szewczak@wp.pl
Coauthor: Boaz Tsaban
A topological space X is Menger if for every sequence of open covers
OS
. . . there are finite sets F1 ⊆ O1 , F2 ⊆ O2 , . . . such that the family
1 , O2 , S
{ F1 , F2 , . . . } is a cover
S of X. If we can request that for every element
x ∈ X the set { n : x ∈ Fn } is co-finite, then the space X is Hurewicz.
The above properties generalize σ-compactness and Hurewicz’s property is
strictly stronger than Menger’s (Chaber-Pol and Tsaban-Zdomskyy).
One of the major open problems in the field of selection principles is
whether there are, in ZFC, two Menger sets of real numbers whose product
is not Menger. We provide examples under various set theoretic hypotheses,
some being weak portions of the Continuum Hypothesis, and some violating
it. The proof method is new.
We also consider filter versions of the above-mentioned properties, and
prove that they are strictly inbetween Hurewicz and Menger.
∗
Supported by Polish National Science Center, UMO-2014/12/T/ST1/00627
project: GO-Spaces and Paracompactness in Cartesian Products
Nano Topology induced by an Ideal
1 M.
1,2 School
Lellis Thivagar
2 V.Sutha
Devi
of Mathematics, Madurai Kamaraj University,
Madurai - 625 021, Tamilnadu, India.
1 E-mail : mlthivagar@yahoo.co.in
2 E-mail : vsdsutha@yahoo.co.in
Abstract
The concept of ideal in topological space was first introduced by Kuratowski. Lellis
Thivagar et al interjected a nano topological space with respect to a subset X of an universe which is defined in terms of lower and upper approximations of X. The elements of
a nano topological space are called the nano-open sets. The topology recommended here
is named so because of its size, Since it has atmost five elements in it. In this paper, a
new definition of lower and upper approximations via ideal have been introduced based
on both any binary reflexive relation and equivalence relation. These new definitions are
compared with nano approximations. It’s therefore shown that the current definitions
are more general. It’s apparent that the present method decreases the boundary region
and we get a topology finer than existing one.
Keywords: Nano topology, binary reflexive relation, Nano Accuracy
2010 AMS SUBJECT CLASSIFICATION: 54B05, 54C05
References
[1] Abd El-Monem., Kozae., On Topology expansions by ideals and applications,
Chaos,Solutions and Fractals., Vol.14 (2002), 619-625.
[2] Hamlett .T.R. and Jankovic.D., Ideals in General Topology, Lecture notes in Pure
and Appl. Math., 123(1990), 115-125.
[3] Kuratowski.K, Topology, Vol.1, Academic Press, Newyork, (1996).
[4] Lellis Thivagar.M and Carmel Richard., On Nano Forms of Weakly Open sets,
Internat.j.Math.and stat.Inv., Vol.1,No.1 (2013), 31-37.
[5] Pawlak .Z., Rough Set theory and its applications, Journal of Telecommunications
and Information Technology, 3, (2002),(7-10).
1
Pseudoradial spaces and the weak Whyburn
property. ∗†
Angelo Bella and Gino Tironi, Emeritus
Department of Mathematics, University of Catania and
Department of Mathematics and Geosciences - University of Trieste,
Section Mathematics ‡
July 21, 2015
Essentially pseudoradial and strongly pseudoradial spaces satisfy the weak
Whyburn property. Here the relation among the two classes of pseudoradial
spaces is examined.
∗ 2010
Mathematics Subject Classification: Primary 54A20; Secondary 54A25.
words and phrases : Pseudoradial spaces, strongly pseudoradial spaces, weakly Whyburn spaces, compact spaces.
‡ Author’s e-mail: bella@unict.it and tironi@units.it
† Key
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Joint metrizability of compact subsets of function spaces
Vladimir V. Tkachuk
Universidad Autonoma Metropolitana de Mexico
Mexico City, Mexico
A space Y is jointly metrizable on compacta (or is a JCM space) if there
exists a metric d on Y such that the topology generated by d restricted to K
coincides with the subspace topology on K whenever K is a compact subspace
of Y . The JCM property was introduced by Arhangel’skii and Al Shumrani
in 2012 and studied systematically by Arhangel’skii and Choban in 2013-2014.
Since every submetrizable space is JCM, this notion generalizes the concept of
submetrizability.
We will present some results on the JCM property in spaces Cp (X). We will
show, among other things, that if Cp (X) is jointly metrizable on compacta, then
p(X) ≤ ω but ω1 need not be a caliber of X. If X is either submetrizable or a
P -space, then Cp (Cp (X)) is jointly metrizable on compacta and, in particular,
all compact subsets of Cp (Cp (X)) are metrizable. We show that for any dyadic
compact X, the space Cp (X) is jointly metrizable on compacta. Therefore,
the JCM property of Cp (X) for a compact space X does not imply that X is
separable. This solves a question published by Arhangel’skii, Choban and Al
Shumrani. If X is a compact space of countable tightness and Cp (X) is jointly
metrizable on compacta, then it is independent of ZFC whether X must be
separable.
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ALGEBRA, SELECTIONS, AND ADDITIVE RAMSEY
THEORY
BOAZ TSABAN
Hindman’s celebrated Finite Sums Theorem asserts that, for each
finite coloring of the set N of natural numbers, there is an infinite set
A ⊆ N such that all finite sums of elements of A have the same color.
This theorem can be viewed as a coloring property of a certain cover
of the set N. We extend this theorem from N to arbitrary topological
spaces with Menger’s classic covering property. The methods include,
in addition to Hurewicz’s game theoretic characterization of Menger’s
property, extensions of the classic idempotent theory in the Stone–
Čech compactification of semigroups, and of the more recent theory
of selection principles. The resulting monochromatic substructures are
large, beyond infinitude, in an analytic sense.
The main results, modulo technical refinements, are of the following
type: Let X be a Menger space, and U be an infinite open cover of
X. For each finite coloring of the family of open subsets of X, there
are
subsets F1 , F2 , . . . of the cover U whose unions V1 :=
S disjoint finite
S
F1 , V2 := F2 , . . . have the following properties:
S
(1) All sets n∈F Vn , for nonempty finite sets F , are of the same
color.
(2) The family {V1 , V2 , . . . } is an open cover of X.
The high-dimensional version of the Finite Sums Theorem, due to
Milliken and Taylor, also extends to Menger spaces in a similar manner.
We will survey the relevant definitions, methods, and results. The
lecture is based on a recent paper available at arxiv.org/abs/1407.7437.
Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002,
Israel.
E-mail address: tsaban@math.biu.ac.il
URL: math.biu.ac.il/~tsaban
MENGER REMAINDERS OF TOPOLOGICAL GROUPS
LYUBOMYR ZDOMSKYY
We will discuss what kind of constrains combinatorial covering properties
of Menger, Scheepers, and Hurewicz impose on remainders of topological
groups. For instance, we show that such a remainder is Hurewicz if and
only it is σ-compact. Also, the existence of a Scheepers non-σ-compact
remainder of a topological group follows from CH and yields a P -point,
and hence is independent of ZFC. We also make an attempt to prove a
dichotomy for the Menger property of remainders of topological groups in
the style of Arhangelskii.
The talk will be based on a joint work with Angelo Bella and Secil Tokgöz
available at
http://arxiv.org/pdf/1504.01626.pdf
Kurt Gödel Research Center for Mathematical Logic, University of
Vienna, Währinger Straße 25, A-1090 Wien, Austria.
E-mail address: lzdomsky@gmail.com
URL: http://www.logic.univie.ac.at/~lzdomsky/
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