Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1016/j.topol.2019.106943

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© 2019 Elsevier B.V. Recall that a space X is selectively pseudocompact if for every sequence {Un:n∈N} of non-empty open subsets of X one can choose a point xn∈Un for all n∈N such that the resulting sequence {xn:n∈N} has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by García-Ferreira and Ortiz-Castillo; the present name is due to Dorantes-Aldama and the first listed author. In 2015, García-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.

DOI： 10.1016/j.topol.2019.06.018

Scopus

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© 2019 Elsevier B.V. We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g∈G can be represented as a product g=g 1 g 2 …g k , where k is a positive integer (depending on g) and the cyclic group generated by each g i is contained in U. In particular, G admits a Hausdorff group topology with the small subgroup generating property of Gould. This provides a positive answer to a question of Comfort and Gould in the case of free groups with infinitely many generators. The case of free groups with finitely many generators remains open.

DOI： 10.1016/j.topol.2019.02.043

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© Instytut Matematyczny PAN, 2019 For a subset A of a group G, we denote by hAi the smallest subgroup of G containing A and let Cyc(A) = {x ∈ G : h{x}i ⊆ A}. A topological group G is SSGP if hCyc(U)i is dense in G for every neighbourhood U of the identity of G. The SSGP groups form a proper subclass of the class of minimally almost periodic groups. Comfort and Gould asked about a characterization of abelian groups which admit an SSGP group topology. An “almost complete” characterization was found by Dikranjan and the first author. The remaining case is resolved here. As a corollary, we give a positive answer to another question of Comfort and Gould by showing that if an abelian group admits an SSGP(n) group topology for some positive integer n, then it admits an SSGP group topology as well.

DOI： 10.4064/fm463-3-2018

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© 2018 by the authors. We give a "naive" (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following "selective" compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence (Un) of non-empty open subsets of G, one can choose a point xn ∈ Un for all n ∈ N in such a way that the resulting sequence (xn) has a p-limit in G; that is, n ∈ N: xn ∈ V ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo-w-bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⊕i∈I Xi, where each space Xi is either maximal or discrete, contains no infinite separable pseudocompact subsets.

DOI： 10.3390/axioms7040086

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Academic Press Inc.  

Dugundji spaces were introduced by Pełczyński as compact Hausdorff spaces X such that every embedding of X into a Tychonoff cube [0,1]A admits a linear extension operator u:C(X)→C([0,1]A) such that ‖u‖=1 and u(1X)=1[0,1]A , where 1X is the constant function on X taking value 1. In this paper we show that a compact space X is Dugundji provided that there exists a linear extension operator u:C(X)→C([0,1]A) satisfying one of the following conditions: (a) ‖u‖&lt
2 and |u(f⋅g)|≤‖g‖⋅|u(|f|)| for all f,g∈C(X)
(b) ‖u‖=1.

DOI： 10.1016/j.jmaa.2018.06.030

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Walter de Gruyter GmbH  

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The "compact-like" properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete. (iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups. (iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

DOI： 10.1515/forum-2017-0010

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Orderability, weak orderability and the existence of continuous weak selections on spaces with a single non-isolated point and their products are discussed. We prove that a closed continuous image X of a suborderable space must be hereditarily paracompact provided that its product X x Y with some non-discrete space Y has a separately continuous weak selection. (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2017.09.030

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arXiv

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

A space X is selectively sequentially pseudocompact if for every family {U-n : n is an element of N} of non-empty open subsets of X, one can choose a point x(n) is an element of U-n for every n is an element of N in such a way that the sequence {x(n) : n is an element of N} has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of Garcia-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of Garcia-Ferreira and Tomita. (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2017.08.020

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER GMBH  

Let I be an infinite set, let {G(i) : i is an element of I} be a family of (topological) groups and let G = Pi(i is an element of J) Gi be its direct product. For J subset of I, p(J) : G Pi(i is an element of J) Gj denotes the projection. We say that a subgroup H of G is
(i) uniformly controllable in G provided that for every finite set J subset of I there exists a finite set K subset of I such that pJ(H) = p(J)(H boolean AND circle plus (i is an element of K) G(i)),
(ii) controllable in G provided that pJ(H) = p(J)(H boolean AND circle plus (i is an element of K) G(i)) for every finite set J subset of I,
(iii) weakly controllable in G if H boolean AND circle plus (i is an element of I) G(i) is dense in H, when G is equipped with the Tychonoff product topology.
One easily proves that ( i) double right arrow ( ii) double right arrow ( iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrowcan be reversed when H is compact, but the second arrow cannot be reversed even when H is compact. Both arrows can be reversed if all groups G(i) are finite. When G(i) = A for all i is an element of I, where A is an abelian group, we show that the first arrow can be reversed for all subgroups H of G if and only if A is finitely generated. We also describe compact groups topologically isomorphic to a direct product of countably many cyclic groups. Connections with coding theory are highlighted.

DOI： 10.1515/forum-2016-0047

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We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces C-p (X, G) of G-valued continuous functions on a space X with the topology of pointwise convergence, for a separable metric group G. A space X is weakly pseudocompact if it is Go-dense in at least one of its compactifications. A topological group G is precompact if it is topologically isomorphic to a subgroup of a compact group. We prove that every weakly pseudocompact precompact topological group is pseudocompact, thereby answering positively a question of M. Tkachenko. (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2017.04.012

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

We prove that the group G = Hom(Z(N), Z) of all homomorphisms from the Baer-Specker group Z(N) to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact sub-sets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product Z(c) of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Nunez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-) free discrete abelian groups need not be reflexive.

DOI： 10.1090/proc/13532

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We say that a topological space X is selectively sequentially pseudocompact if for every family {U-n : n is an element of N} of non-empty open subsets of X, one can choose a point x(n) is an element of U-n for every n is an element of N in such a way that the sequence {x(n) : n is an element of N} has a convergent subsequence. We show that the class of selectively sequentially pseudocompact spaces is closed under arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garcia-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every omega-bounded (= the closure of each countable set is compact) group is selectively sequentially pseudocompact, while compact spaces need not be selectively sequentially pseudocompact. Finally, we construct selectively sequentially pseudocompact group topologies on both the free group and the free Abelian group with continuum-many generators. (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2017.02.016

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We call a subset A of an abelian topological group G: (i) absolutely Cauchy summable provided that for every open neighbourhood U of 0 one can find a finite set F subset of A such that the subgroup generated by A\F is contained in U; (ii) absolutely summable if, for every family {z(a) : a is an element of A} of integer numbers, there exists g is an element of G such that the net {Sigma(a is an element of F) z(a)a : F subset of A is finite} converges to g; (iii) topologically independent provided that 0 is not an element of A and for every neighbourhood W of 0 there exists a neighbourhood V of 0 such that, for every finite set F subset of A and each set {z(a) : a is an element of F} of integers, Sigma(a is an element of F) z(a)a is an element of V implies that z(a)a is an element of W for all a is an element of F. We prove that: (1) an abelian topological group contains a direct product (direct sum) of kappa-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality kappa; (2) a topological vector space contains R-(N) as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains R-N as its subspace if and only if it has an R-N multiplier convergent series of non-zero elements. We answer a question of Husek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki. (C) 2016 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2016.01.037

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We introduce and study a functorial topology on every group G having as a base the family of all subgroups of G. Making use of this topology, we obtain an equivalent description of the small subgroup generating property introduced by Gould [26]; see also Comfort and Gould [6]. This property implies minimal almost periodicity. Answering questions of Comfort and Gould [6], we show that every abelian group of infinite divisible rank admits a group topology having the small subgroup generating property. For unbounded abelian groups of finite divisible rank, we find a new necessary condition for the existence of a group topology having the small subgroup generating property, and we conjecture that this condition is also sufficient. (C) 2015 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2015.12.015

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Every proper closed subgroup of a connected Hausdorff group must have index at least c, the cardinality of the continuum. 70 years ago Markov conjectured that a group G can be equipped with a connected Hausdorff group topology provided that every subgroup of G which is closed in all Hausdorff group topologies on G has index at least c. Counter-examples in the non-abelian case were provided 25 years ago by Pestov and Remus, yet the problem whether Markov's Conjecture holds for abelian groups G remained open. We resolve this problem in the positive. (C) 2015 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2015.09.006

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN  

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism (G) over cap -> (D) over cap of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G(delta)-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its G(delta)-dense subgroups is metrizable, thereby resolving a question of Hernandez, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Dominguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building G(delta)-dense subgroups without uncountable compact subsets in compact groups of weight omega(1) (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.

DOI： 10.4064/fm221-2-3

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The classical result of Landau on the existence of kings in finite tournaments (= finite directed complete graphs) is extended to continuous tournaments for which the set X of players is a compact Hausdorff space. The following partial converse is proved as well. Let X be a Tychonoff space which is either zero-dimensional or locally connected or pseudocompact or linearly ordered. If X admits at least one continuous tournament and each continuous tournament on X has a king, then X must be compact. We show that a complete reversal of our theorem is impossible, by giving an example of a dense connected subspace Y of the unit square admitting precisely two continuous tournaments both of which have a king, yet Y is not even analytic (much less compact). (C) 2012 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2012.05.021

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-V C H VERLAG GMBH  

Let G be an abelian topological group. The symbol (G) over cap denotes the group of all continuous characters. : G. T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that chi(E) subset of phi ([-1/4, 1/4]) holds only for the trivial character chi is an element of (G) over cap, where phi : R -> T = R/Z is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most one non-isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G(a) contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Aussenhofer: For a connected locally compact abelian group G, the restriction homomorphism r : (G) over cap -> (G) over cap (a) G(a) defined by r (chi) = chi vertical bar G(a) for chi is an element of (G) over cap, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component G(a) contains a super-sequence converging to the identity that is qc-dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

DOI： 10.1002/mana.201010013

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2(c) and a countable family E of infinite subsets of G. we construct "Baire many" monomorphisms pi : G -> T(c) such that pi(E) is dense in {y is an element of T(c) : ny = 0} whenever n is an element of N, E is an element of E, nE = {0} and {x is an element of E: mx = g} is finite for all g is an element of G and m is an element of N\{0} such that n = mk for some k is an element of N\{1}. We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko (2002) [22, Problem 6.5]. Applications to group actions and discrete flows on T(c), Diophantine approximation, Bohr topologies and Bohr compactifications are also provided. (C) 2010 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2010.12.016

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arXiv

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Given a function f : N -> (omega + 1)\{0}, we say that a faithfully indexed sequence {a(n): n is an element of N} of elements of a topological group G is: (i) f-Cauchy productive ( f-productive) provided that the sequence {Pi(m)(n=0)a(n)(z(n)): m is an element of N} is left Cauchy (converges to some element of G, respectively) for each function z : N -> Z such that vertical bar z(n)vertical bar <= f(n) for every n is an element of N; (ii) unconditionally f-Cauchy productive (unconditionally f-productive) provided that the sequence {a(phi(n)): n is an element of N} is (f circle phi)-Cauchy productive (respectively, (f circle phi)-productive) for every bijection phi : N -> N. (Bijections can be replaced by injections here.) We consider the question of existence of (unconditionally) f-productive sequences for a given "weight function" f. We prove that: (1) a Hausdorff group having an f-productive sequence for some f contains a homeomorphic copy of the Cantor set; (2) if a non-discrete group is either locally compact Hausdorff or Weil complete metric, then it contains an unconditionally f-productive sequence for every function f : N -> N \{0}; (3) a metric group is NSS if and only if it does not contain an f(omega)-Cauchy productive sequence, where f(omega) is the function taking the constant value omega. We give an example of an f(omega)-productive sequence {a(n): n is an element of N} in a (necessarily non-abelian) separable metric group H with a linear topology and a bijection phi : N -> N such that the sequence {Pi(m)(n=0)a(phi(n)): m is an element of N} diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that H has no unconditionally f(omega)-productive sequences. As an application of our results, we resolve negatively a question from C-p(-, G)-theory. (C) 2010 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2010.11.009

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

According to Markov (1946) 1241, a subset of an abelian group G of the form (x is an element of G: nx = a), for some integer n and some element a is an element of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally buounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian T(1) topology 3(G) on G called the Zariski, or verbal, topology of G; see Bryant (1977) [31. We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Frechet-Urysohn.
For a countable family 3 of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology 'T on G such that the T-closure of each member of g coincides with its 3(G)-closure. As an application, we provide a characterization of the subsets of G that are 'Tdense in some Hausdorff group topology T on C. and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a longstanding problem of Markov (1946)124]. (C) 2010 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2010.04.025

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arXiv

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

For an uncountable cardinal tau and a subset S of an abelian group G, the following conditions are equivalent:
(i) vertical bar{ns : s is an element of S}vertical bar >= tau for all integers n >= 1;
(ii) there exists a group homomorphism pi : G -> T(2 tau) such that pi(S) is dense in T(2 tau).
Moreover, if vertical bar G vertical bar <= 2(2 tau), then the following item can be added to this list:
(iii) there exists an isomorphism pi : G -> G' between G and a subgroup G' of T(2 tau) such that pi(S) is dense in T(2 tau).
We prove that the following conditions are equivalent for an uncountable subset S of an abelian group G that is either (almost) torsion-free or divisible:
(a) S is T-dense in G for some Hausdorff group topology T on G;
(b) S is T-dense in some precompact Hausdorff group topology T on G;
(c) vertical bar{ns : s is an element of S}vertical bar >= min {tau : vertical bar G vertical bar <= 2(2 tau)} for every integer n >= 1.
This partially resolves a question of Markov going back to 1946.

DOI： 10.1090/S0002-9939-10-10302-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Let G be a topological group with the identity element e Given a space X, we denote by COX G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence. and we say that X is (a) G-regular if, for each closed set F subset of X and every point x is an element of X \ F, there exist f is an element of C(p)(X G) and g is an element of G \ {e} such that f(x) = g and f (F} subset of {e}, (b) G* -regular provided that there exists g is an element of G \ {e} such that, for each closed set F subset of X and every point x is an element of X \ F, one can find f is an element of C(p)(X G) With f (x) - g and f (F) subset of {e} Spaces X and Y are G-equivalent provided that the topological groups C(p) (X, G) and C(p)(Y G) are topologically isomorphic.
We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of C(p)(X,G) Since -equivalence coincides with I-equivalence, this line of research "includes" major topics of the classical C(p)-theory of Arhangel' skill as a particular case (when G = R) We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups) We prove that (i) for a given NSS group C. a G-regular space X is pseudocompact if and only if C(p)(X G) is TAP, and (n) for a metrizable NSS group G, a G*-regular space X is compact if and only if C(p)(X, G) is a TAp group of countable tightness In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X R) is a TAP group (of countable tightness) Demonstrating the limits of the result in (1), we give an example of a precompact TAP group G and a G-regular countably compact space X such that C(p)(X, G) is not TAP
We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G We establish that T-equivalence preserves the following topological properties compactness, pseudocompactness, sigma-compactness. the property of being a Lindelof Sigma-space. the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed (C) 2009 Elsevier B V All rights reserved

DOI： 10.1016/j.topol.2009.06.022

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

For an abelian topological group G, let (G) over cap denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G), and an open neighborhood U of 0 in T, we show that vertical bar{chi is an element of <(G)over cap>: chi(X) subset of U}vertical bar = vertical bar(G) over cap vertical bar. (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the map r: (G) over cap -> (D) over cap defined by r(chi) = chi (sic)D for chi is an element of(G) over cap, is an isomorphism between (G) over cap and (D) over cap. We prove that
w(G) = min{vertical bar D vertical bar: D is a subgroup of G that determines G}
for every infinite compact abelian group G. In particular, an infinite compact abelian group determined by a countable subgroup is metrizable. This gives a negative answer to a question of Comfort, Raczkowski and Trigos-Arrieta (repeated by Hernandez, Macario and Trigos-Arrieta). As an application, we furnish a short elementary proof of the result from [S. Hernandez, S. Macario, FJ. Trigos-Arrieta, Uncountable products of determined groups need not be determined, J. Math. Anal. Appl. 348 (2008) 834-842] that a compact abelian group G is metrizable provided that every dense subgroup of G determines G. (C) 2009 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2009.07.038

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A Hausdorff topological group G is minimal if every continuous isomorphism f : G -> H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence {sigma(n): n is an element of N} of cardinals such that
w(G) = sup{sigma(n): n is an element of N} and sup{2(sigma n): n is an element of N} <= vertical bar G vertical bar <= 2(w(G)).
where w(G) is the weight of G. If G is an infinite minimal abelian group, then either vertical bar G vertical bar = 2(sigma) for some cardinal sigma, or w(G) = min{sigma: vertical bar G vertical bar <= 2(sigma)}: moreover, the equality vertical bar G vertical bar = 2(w(G)) holds whenever cf(w(G)) > omega.
For a cardinal kappa, we denote by F(kappa) the free abelian group with kappa many generators, if F(kappa) admits a pseudocompact group topology, then kappa >= c, where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology oil F(c) is equivalent to the Lusin's Hypothesis 2(omega 1) = c. For kappa > c, we prove that F(kappa) admits a (zero-dimensional) minimal pseudocompact group topology if and only if F(kappa) has both a minimal group topology and a pseudocompact group topology. If K > C, then F, admits a connected minimal pseudocompact group topology of weight sigma if and only if kappa = 2(sigma). Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology. (C) 2009 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2009.03.028

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If a discrete subset S of a topological group G with the identity I generates a dense subgroup of G and S boolean OR (1) is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory. (C) 2008 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2008.12.009

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER & CO  

We give a necessary and sufficient condition, in terms of a certain :reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63-year-old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G. Essential connections with non-topologizable groups are highlighted.

DOI： 10.1515/JGT.2008.025

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-V C H VERLAG GMBH  

Let G be a compact Hausdorff group. A subspace X of G topologically generates G if G is the smallest closed subgroup of G containing X. Define
tgw(G) = w - min{w(X) : X is closed in G and topologically generates G},
where w(X) is the weight of X, i.e., the smallest size of a base for the topology of X. We prove that: (i) tgw (G) = w(G) if G is totally disconnected, (ii) tgw(G) = w root w(G) = min{tau >= w : w(G) <= tau(w)} in case G is connected, and (iii) tgw(G) = w(G/c(G)) . w root w(c(G)), where c(G) is the connected component of G.
If G is connected, then either tgw (G) = w(G), or cf (tgw(G)) = w (and, moreover, w(G) = tgw(G)(+) under the Singular Cardinal Hypothesis).
We also prove that tgw(G) = w . min{vertical bar X vertical bar : X subset of G is a compact Hausdorff space with at most one non-isolated point topologically generating G}. (c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

DOI： 10.1002/mana.200410499

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Language：English   Publishing type：Part of collection (book)   Publisher：Elsevier  

This chapter discusses selected topics from the structure theory of topological groups. It contains open problems and questions covering the a number of topics including: the dimension theory of topological groups, pseudocompact and countably compact group topologies on Abelian groups, with or without nontrivial convergent sequences, categorically compact groups, sequentially complete groups, the Markov-Zariski topology, the Bohr topology, and transversal group topologies. All topological groups considered in this chapter are assumed to be Hausdorff. It is stated that Abelian group G is algebraically compact provided that an Abelian group H is found such that G ×. H admits a compact group topology. Algebraically compact groups form a relatively narrow subclass of Abelian groups (for example, the group ℤ of integers is not algebraically compact). On the other hand, every Abelian group G is algebraically pseudo-compact
that is, an Abelian group H can be found such that G ×. H ∈. P. Problems and questions related to Bohr-homeomorphic bounded Abelian groups are also discussed in the chapter. © 2007 Elsevier B.V. All rights reserved.

DOI： 10.1016/B978-044452208-5/50041-7

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2(c) "arbitrarily large" and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2(c)) which admit:
(i) a hereditarily separable group topology,
(ii) a group topology making G into an S-space,
(iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets),
(iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary).
As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most 2(c) which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired).
We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups. (c) 2004 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2004.07.012

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Given a space Y, let us say that a space X is a total extender for Y provided-that every continuous map f : A --> Y defined on a subspace A of X admits a continuous extension (f) over tilde : X --> Y over X. The first author and Alberto Marcone proved that a space X is hereditarily extremally disconnected and hereditarily normal if and only if it is a total extender for every compact metrizable space Y, and asked whether the same result holds without any assumption of metrizability on Y. We demonstrate that a hereditarily extremally disconnected, hereditarily normal, non-collectionwise Hausdorff space X constructed by Kenneth Kunen is not a total extender for K, the one-point compactification of the discrete space of size omega(1). Under the assumption 2(omega0) = 2(omega1), we provide an example of a separable, hereditarily extremally disconnected, hereditarily normal space X that is not a total extender for K. Furthermore, using forcing we prove that, in the generic extension of a model of ZFC + MA(omega(1)), every first-countable separable space X of size omega(1) has a finer topology tau on X such that (X, tau) is still separable and fails to be a total extender for K. We also show that a hereditarily extremally disconnected, hereditarily separable space X satisfying some stronger form of hereditary normality (so-called structural normality) is a total extender for every compact Hausdorff space, and we give a non-trivial example of such an X. (C) 2004 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2004.02.016

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV HOUSTON  

A pair tau(1), tau(2) of T-1 topologies on an infinite set X is called T-1-independent if their intersection tau(1) boolean AND tau(2) is the cofinite topology, and transversal if the union tau(1) boolean OR tau(2) generates the discrete topology. We show that every Hausdorff space admits a transversal compact Hausdorff topology. Then we apply Booth's Lemma to prove that no infinite set of cardinality less than 2(omega) admits a pair of T-1-independent Hausdorff topologies. This answers, in a strong form, a question posed by S. Watson in 1996. It is shown in ZFC that betaomega\omega is a self T-1-independent compact Hausdorff space, but the existence of self T-1-independent compact Hausdorff spaces of cardinality 2(omega) is both consistent with and independent of ZFC.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

Recall that a topological group G is: ( a) sigma-compact if G = boolean OR{K-n : n epsilon N} where each K-n is compact, and ( b) compactly generated if G is algebraically generated by some compact subset of G. Compactly generated groups are sigma - compact, but the converse is not true: every countable nonfi nitely generated discrete group ( for example, the group of rational numbers or the free ( Abelian) group with a countable in finite set of generators) is a counterexample. We prove that a metric group G is compactly generated if and only if G is sigma - compact and for every open subgroup H of G there exists a finite set F such that F boolean OR H algebraically generates G. As a corollary, we obtain that a sigma - compact metric group G is compactly generated provided that one of the following conditions holds: ( i) G has no proper open subgroups, ( ii) G is dense in some connected group ( in particular, if G is connected itself), ( iii) G is totally bounded (= subgroup of a compact group). Our second major result states that a countable metric group is compactly generated if and only if it can be generated by a sequence converging to its identity element ( eventually constant sequences are not excluded here). Therefore, a countable metric group G can be generated by a ( possibly eventually constant) sequence converging to its identity element in each of the cases ( i), ( ii) and ( iii) above. Examples demonstrating that various conditions cannot be omitted or relaxed are constructed. In particular, we exhibit a countable totally bounded group which is not compactly generated.

DOI： 10.1090/S0002-9939-02-06736-9

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS  

Topologies tau(1) and tau(2) on a set X are called T-1-complementary tau(1) boolean AND tau(2) = {X \ F : F subset of X is finite} boolean OR {0} and tau(1) boolean OR tau(2) is a subbase for the discrete topology on X. Topological spaces (X, tau(X)) and (Y, tau(Y)) are called T-1-complementary provided that there exists a bijection f : X --> Y such that tau(X) and {f(-1)(U) : U is an element of tau(Y)} are T-1-complementary topologies on X. We provide an example of a compact Hausdorff space of size 2(c) which is T-1-complementary to itself (c denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff space of size c that is T-1-complementary to itself is both consistent with and independent of ZFC. On the other hand, we construct in ZFC a countably compact Tikhonov space of size c which is T-1-complementary to itself and a compact Hausdorff space of size c which is T-1-complementary to a countably compact Tikhonov space. The last two examples have the smallest possible size: It is consistent with ZFC that c is the smallest cardinality of an infinite set admitting two Hausdorff T-1-complementary topologies [8]. Our results provide complete solutions to Problems 160 and 161 (both posed by S. Watson [14]) from Open Problems in Topology (North-Holland, 1990).

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

For a topological space X let Z(sigma)(X) denote the family of subsets of X which can be represented as a union of countably many zero-sets. A bijection h :X --> Y between topological spaces X and Y is a first level Baire isomorphism if f(Z) is an element of Z(sigma)(Y) and f(-1)(Z') is an element of Z(sigma)(X) whenever Z is an element of Z(sigma)(X) and Z' is an element of Z(sigma)(Y) A space is sigma-(pseudo)compact if it can be represented as the union of a countable family consisting of its (pseudo)compact subsets. Generalizing results of Jayne, Rogers and Chigogidze we show that first level Baire isomorphic, sigma-pseudocompact (in particular, sigma-compact) Tychonoff spaces have the same covering dimension dim. (C) 2000 Elsevier Science B.V. All rights reserved.

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It is natural to expect that the behaviour of some topological properties tends to improve in the presence of an additional algebraic structure interacting with the topology (for example, in topological groups, topological fields, or topological vector spaces). The purpose of this survey is to compare topological groups, topological vector spaces and topological fields as to how far each of these classes of spaces is from the class of Tychonoff spaces. In other words, we want to compare the degree of how much of an additional strain an algebraic structure of a group, vector space or field which agrees with the topology of the space imposes on the topology of that space. We cover selected results and open problems related to normality-type properties, covering properties, Cartesian products, homeomorphic embeddings and dimension theory. (C) 1999 Elsevier Science B.V. All rights reserved.

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For a category K we use Ob(K) to denote the class of all objects of K:, if X, Y is an element of Ob(K), then Mor(K)(X, Y) is the set of all K-morphisms from X into Y. Let PA and a be subcategories of the category of all topological spaces and their continuous maps. We say that a covariant functor F:A --> B is an embedding functor if there exists a class (i(X): X is an element of Ob(A)} satisfying the following conditions: (i) i(x) :X --> F(X) is a homeomorphic embedding for every X is an element of Ob(A), and (ii) if X, Y is an element of Ob(A) and f is an element of Mor(K)(X, Y), then F(S) o i(X) = i(Y) o f. For a natural number n let C(n) denote the category of all n-dimensional compact metric spaces and their continuous maps. Let G(< infinity) be the category of all Hausdorff finite-dimensional topological groups and their continuous group homomorphisms. We prove that there is no embedding covariant functor F:C(1) --> G(< infinity), but there exists a covariant embedding functor F:C(0) --> G(0), where G(0) is the category consisting of the single (zero-dimensional) compact metric group Z(2)(omega) and all its continuous group homomorphisms into ifself i.e., Ob(G(0)) = {Z(2)(w)) and MorG((0))(Z(2)(omega),Z(2)(omega)) is the set of all continuous group homomorphisms from Z(2)(omega) into Z(2)(omega). (C) 1998 Elsevier Science B.V.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

Looking for a natural generalization of compact spaces, in 1948 Hewitt introduced pseudocompact spaces as those Tychonoff spaces on which every real-valued continuous function is bounded. The algebraic structure of compact Abelian groups was completely described in the fifties and sixties by Kaplansky, Harrison and Hulanicki. In this paper we study systematically the algebraic structure of pseudocompact groups, or equivalently, the following problem: Which groups can be equipped with a pseudocompact topology turning them into topological groups? We solved this problem completely for the following classes of groups: free groups and free Abelian groups (or more generally, free groups in some variety of abstract groups), torsionfree Abelian groups (or even Abelian groups G with \G\ = r(G)), torsion Abelian groups, and divisible Abelian groups.
Even though out main problem deals with the existence of some topologies on groups, it has a strong set-theoretic flavor. Indeed, the existence of an infinite pseudocompact group of cardinality tau and weight sigma is equivalent to the following purely set-theoretic condition Ps(tau, sigma) introduced by Cater, Erdos and Galvin for entirely different purposes: The set {0, 1)(sigma) of all functions from (a set of cardinality) a to the two-point set {0, 1} contains a subset of size tau whose projection on every countable subproduct {0, 1}(A) is a surjection. Despite its innocent look, the problem of which cardinals sigma and tau enjoy such a relationship is far from being solved, and is closely related to the Singular Cardinal Hypothesis.
A variety of necessary conditions, both of algebraic and of set-theoretic nature, for the existence of a pseudocompact group topology on a group is discovered. For example, pseudocompact torsion groups are locally finite. If an infinite Abelian group G admits a pseudocompact group topology of weight sigma, then either r(p)(r(G),a) or Ps(rp(G),a) for some prime number p must hold, where r(G) and rp(G) are the free rank and the prank of G respectively. If an Abelian group G has a pseudocompact group topology, then \{ng : g is an element of G}\ less than or equal to 2(2r(G)) for some n. This yields the inequality \G\ less than or equal to 2(2r(G)) for a divisible pseudocompact group.
Turning to necessary and sufficient conditions, we show that a nontrivial Abelian group G admits a connected pseudocompact group topology of weight a if and only if \G\ less than or equal to 2(sigma) and Ps(r(G),sigma) hold. Moreover, a free group with tau generators in a variety nu of groups admits a pseudocompact group topology if and only if Ps(tau, sigma) holds for some infinite sigma, and the variety nu is generated by its finite groups. It should be noted that most of the classical varieties of groups have the last property, the only exception the authors are aware of being the Burnside varieties B-n for odd n > 665.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

A connected Tychonoff space X is called maximal Tychonoff connected if there is no strictly finer Tychonoff connected topology on X. We show that if X is a connected Tychonoff space and X is an element of {locally separable spaces, locally Cech-complete spaces, first countable spaces}, then X is not maximal Tychonoff connected. This result is new even in the cases where X is compact or metrizable.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIONE MAT ITAL  

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We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long line L, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological space X is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a) X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection
(b) X is infinite, separable, locally connected and has exactly one continuous selection
(c) X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results. © 1997 Springer.

DOI： 10.1007/BF02977032

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We characterize Lindelof p-spaces which are absolute extensors for zero-dimensional perfectly normal spaces. As an application we prove that a Lindelof Cech-complete space X is an absolute extensor for zero-dimensional spaces if and only if there exists an upper semi-continuous compact-valued map r : X(3) -> X such that r(x, y, y) = r(y, y, x) = {x} for all x, y is an element of X. This result is new even when applied to compact spaces and yields the following new characterization of Dugundji spaces: A compact Hausdorff space X is Dugundji if and only if there exists an upper semi-continuous compact-valued map r : X(3) -> X such that r(x, y, y) = r(y, y, x) = {x} for all x, y is an element of X. It is worth noting that, by a result of Uspenskij, in the above characterization of Dugundji spaces the set-valued map r cannot be replaced by a single-valued (continuous) map, the 5-dimensional sphere S(5) being a counterexample.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We characterize Lindelof p-spaces which are absolute extensors for zero-dimensional perfectly normal spaces. As an application we prove that a Lindelof Cech-complete space X is an absolute extensor for zero-dimensional spaces if and only if there exists an upper semi-continuous compact-valued map r : X(3) -> X such that r(x, y, y) = r(y, y, x) = {x} for all x, y is an element of X. This result is new even when applied to compact spaces and yields the following new characterization of Dugundji spaces: A compact Hausdorff space X is Dugundji if and only if there exists an upper semi-continuous compact-valued map r : X(3) -> X such that r(x, y, y) = r(y, y, x) = {x} for all x, y is an element of X. It is worth noting that, by a result of Uspenskij, in the above characterization of Dugundji spaces the set-valued map r cannot be replaced by a single-valued (continuous) map, the 5-dimensional sphere S(5) being a counterexample.

DOI： 10.1016/S0166-8641(96)00049-1

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

For a given topological space X we consider two topologies on the hyperspace F(X) of all closed subsets of X. The Fell topology T-F on F(X) is generated by the family {O-VK: V is open in X and K subset of or equal to X is compact} as a subbase, where O-VK = {F is an element of F(X): F boolean AND V not equal 0 and F boolean AND K = 0}. The topology T-F is always compact, regardless of the space X. The Kuratowski topology T-K is the smallest topology on F(X) which contains both the lower Vietoris topology T-iV, generated by the family { { F is an element of F(X): F\Phi not equal 0}: Phi is an element of F(X)} as a subbase, and the upper Kuratowski topology T-uK, which is the strongest topology on F(X) such that upper Kuratowski-Painleve convergence of an arbitrary net of closed subsets of X to some closed set A implies that the same net, considered as a net of points of the topological space (F(X),T-uK), converges in this space to the point A. [Recall that a net [A(lambda)](lambda is an element of Lambda) subset of or equal to F(X) upper Kuratowski-Painleve converges to A if boolean AND{<(boolean OR{A(mu):mu greater than or equal to lambda}:)over bar lambda is an element of Lambda>} subset of or equal to A.] The inclusion T-F subset of or equal to T-K holds for an arbitrary space X, while the equation T-F = T-K is equivalent to consonance of X, the notion recently introduced by Dolecki, Greco and Lechicki. These three authors showed that complete metric spaces are consonant. In our paper we give an example of a metric space with the Faire property which is not consonant. We also demonstrate that consonance is a delicate property by providing an example of two consonant spaces X and Y such that their disjoint union X+Y is not consonant. In particular, locally consonant spaces need not be consonant.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier  

For a given topological space X we consider two topologies on the hyperspace F(X) of all closed subsets of X. The Fell topology ΤF on F(X) is generated by the family {OVK: V is open in X and K ⊆ X is compact} as a subbase, where OVK = {F ∈ F(X): F ∩ V ≠ ø and F ∩ K = ø}. The topology ΤF is always compact, regardless of the space X. The Kuratowski topology ΤK is the smallest topology on F(X) which contains both the lower Vietoris topology ΤIV, generated by the family { { F ∈ F(X): F \\ Φ ≠ ø}: Φ ∈ F(X)} as a subbase, and the upper Kuratowski topology ΤuK, which is the strongest topology on F(X) such that upper Kuratowski-Painlevé convergence of an arbitrary net of closed subsets of X to some closed set A implies that the same net, considered as a net of points of the topological space (F(X), ΤuK), converges in this space to the point A. [Recall that a net 〈Aλ〉λ∈Λ ⊆ F(X) upper Kuratowski-Painlevé converges to A if ∩{∪{Aμ μ ≥ λ}: λ ∈ Λ.] The inclusion ΤF ⊆ ΤK holds for an arbitrary space X, while the equation ΤF = ΤK is equivalent to consonance of X, the notion recently introduced by Dolecki, Greco and Lechicki. These three authors showed that complete metric spaces are consonant. In our paper we give an example of a metric space with the Baire property which is not consonant. We also demonstrate that consonance is a delicate property by providing an example of two consonant spaces X and Y such that their disjoint union X ⊕ Y is not consonant. In particular, locally consonant spaces need not be consonant. © 1996 Elsevier Science B.V. All rights reserved.

DOI： 10.1016/0166-8641(95)00098-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：NEW YORK ACAD SCIENCES  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：GAUTHIER-VILLARS  

Following Arhangel'skii we say that a space X has the alpha(1)-property (alpha(4)-property respectively) if for every countable family {S-n : n is an element of N} of infinite sequences converging to some point x is an element of X there exists a [diagonal] sequence S converging to x such that S-n\S is finite for all n is an element of N (such that S intersects infinitely many S-n respectively). We show that, while different for general topological groups, these two converging properties coincide for locally compact groups, and for such groups are also equivalent to the so-called Ramsey property. We also establish that, under some additional set-theoretic assumption beyond the classical Zermelo-Fraenkel axioms ZFC of set theory, every locally compact group with (any of the two) amalgamation properties mentioned above is metric, and that at least some extra set-theoretic assumption beyond ZFC is necessary for the last result.

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Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：BOLYAI JANOS MATEMATIKA TARSULAT  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN  

We prove that every nonmetrizable compact connected Abelian group G has a family H of size \G\, the maximal size possible, consisting of proper dense pseudocompact subgroups of G such that H boolean AND H' = {0} for distinct H, H' is an element of H. An easy example shows that connectedness of G is essential in the above result. In the general case we establish that every nonmetrizable compact Abelian group G has a family H of size \G\ consisting of proper dense pseudocompact subgroups of G such that each intersection H boolean AND H' of different members of H is nowhere dense in G. Some results in the non-Abelian case are also given.

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DOI： 10.1111/j.1749-6632.1994.tb44152.x

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

A cover C of a topological space X is point-countable (point-finite) if every point of X belongs to at most countably many (at most finitely many) elements of C. We say that a space X has the weak topology with respect to a cover C provided that a set F subset-or-equal-to X is closed in X if and only if its intersection F and C with every C is-an-element-of C is closed in C. A space X is an alpha4-space if for every point x is-an-element-of X and any countable family {S(n): n is-an-element-of N} of sequences converging to x one can find a sequence S converging to x which meets infinitely many S(n).
The classical Birkhoff-Kakutani theorem says that a Hausdorff topological group is metrizable if (and only if) it is first countable. Quite recently Arhangel'skii generalized this theorem by showing that Hausdorff bisequential topological groups are metrizable (recall that first countable spaces are bisequential). In our paper we generalize these results by showing that a Hausdorff topological group is metrizable if it has the weak topology with respect to a point-finite cover consisting of bisequential spaces. In addition we establish the following theorem each item of which also generalizes both Birkhoff-Kakutani's and Arhangel'skii's results: Theorem. Let G be a Hausdorff topological group which has the weak topology with respect to a point-countable cover C consisting of bisequential spaces. Then G is metrizable in each of the following cases: (i) G is an alpha4-space, (ii) C consists of closed subspaces and G does not contain a closed subspace homeomorphic to S(omega) (or equivalently, a closed subspace homeomorphic to S2). (iii) C is countable and increasing, and G contains no closed subspace homeomorphic to S(omega) (equivalently, no closed subspace homeomorphic to S2).
Recall that S(omega) is the quotient space obtained from the union of a countable family of convergent sequences via identifying their limit points, and S2 is Arens' space, the standard sequential space of sequential order 2.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We make few observations about the specific behaviour of transfinite inductive dimensions in topologically homogeneous spaces and topological groups. Two main results, are: (i) If the large transfinite inductive dimension trInd X of a homogeneous normal space X is defined, then either ind X is finite or X is countably compact. (ii) If G is a normal topological group having the large transfinite inductive dimension trInd G, then ind G is finite.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

Denote by G the class of all Abelian Hausdorff topological groups. A group G is-an-element-of G is minimal (totally minimal) if every continuous group isomorphism (homomorphism) i: G --> H of G onto H is-an-element-of G is open. For G is-an-element-of G let K(G) be the smallest cardinal tau greater-than-or-equal-to 1 such that the minimality of G(tau) implies the minimality of all powers of G. For Q subset-of G, Q not-equal phi, we set kappa(Q) = sup{kappa(G): G is-an-element-of G} and denote by alpha(Q) the smallest cardinal tau greater-than-or-equal-to 1 having the following property: If {G(i): i is-an-element-of I} subset-of Q, I not-equal phi, and each subproduct PI{G(i): i is-an-element-of J}, with J subset-of 1, J not-equal phi, and Absolute value of J less-than-or-equal-to tau, is minimal, then the whole product PI{G(i): i is-an-element-of I} is minimal. These definitions are correct, and kappa(G) less-than-or-equal-to 2omega and kappa(Q) less-than-or-equal-to alpha(Q) less-than-or-equal-to 2omega for all G is-an-element-of G and any Q subset-of G, Q not-equal phi, while it can happen that kappa(Q) < alpha(Q) for some Q subset-of G. Let C = {G is-an-element-of G : G is countably compact{ and P = {G is-an-element-of G: G is pseudocompact}. If G is-an-element-of C is minimal, then G x H is minimal for each minimal (not necessarily Abelian) group H ; in particular, G(n) is minimal for every natural number n . We show that alpha(C) = omega, and so either kappa(C) = 1 or kappa(C) = omega. Under Lusin's Hypothesis 2omega1 = 2omega we construct {G(n): n is-an-element-of N} subset-of P and H is-an-element-of P such that: (i) whenever n is-an-element-of N, G(n)n is totally minimal, but G(n)n+1 is not even minimal, so kappa(G(n)) = n+1 ; and (ii) H(n) is totally minimal for each natural number n , but H(omega) is not even minimal, so kappa(H) = omega. Under MA + -CH, conjunction of Martin's Axiom with the negation of the Continuum Hypothesis, we construct G is-an-element-of P such that G(tau) is totally minimal for each T < 2omega, while G2omega is not Minimal, so kappa(G) = 2omega. This yields alpha(P) = kappa(P) = 2omega under MA + -CH. We also present an example of a noncompact minimal group G is-an-element-of C, which should be compared with the following result obtained by the authors quite recently: Totally minimal groups G is-an-element-of C are compact.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

A subgroup H of a topological group G is (weakly) totally dense in G if for each closed (normal) subgroup N of G the set H the-intersection-of N is dense in N. We show that no compact (or more generally, omega-bounded) group contains a proper, totally dense, countably compact subgroup. This yields that a countably compact Abelian group G is compact if and only if each continuous homomorphism pi: G --> H of G onto a topological group H is open. Here "Abelian" cannot be dropped. A connected, compact group contains a proper, weakly totally dense, countably compact subgroup if and only if its center is not a G(delta)-subgroup. If a topological group contains a proper, totally dense, pseudocompact subgroup, then none of its closed, normal G(delta)-subgroups is torsion. Under Lusin's hypothesis 2(omega)1 = 2(omega) the converse is true for a compact Abelian group G. If G is a compact Abelian group with nonmetrizable connected component of zero, then there are a dense, countably compact subgroup K of G and a proper, totally dense subgroup H of G with K is-contained-in-or-equal-to H (in particular, H is pseudocompact).

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AKADEMIAI KIADO  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER VERLAG  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACAD BULGARE DES SCIENCES  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACAD BULGARE DES SCIENCES  

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Language：Russian   Publisher：MOSCOW STATE UNIV  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PLENUM PUBL CORP  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

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Language：Russian   Publisher：MOSCOW STATE UNIV  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PLENUM PUBL CORP  

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Language：Russian   Publishing type：Research paper (scientific journal)   Publisher：MEZHDUNARODNAYA KNIGA  

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Language：Russian   Publishing type：Research paper (scientific journal)   Publisher：MOSCOW STATE UNIV  

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Language：Russian   Publishing type：Research paper (scientific journal)   Publisher：MEZHDUNARODNAYA KNIGA  

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Language：English   Publisher：Kyoto University  

CiNii Books

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Other Link： http://hdl.handle.net/2433/41750

Language：English   Publisher：Kyoto University  

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Other Link： http://hdl.handle.net/2433/64710

Language：English   Publisher：Kyoto University  

CiNii Books

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Other Link： http://hdl.handle.net/2433/63900

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Language：Russian   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)   Publisher：MOSCOW STATE UNIV  

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Language：Russian   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)   Publisher：MOSCOW STATE UNIV  

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Event date： 2019.6

Presentation type：Oral presentation (general)  

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Presentation type：Oral presentation (keynote)  

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Language：English   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (general)  

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