M. Tkachenko | Universidad Autónoma Metropolitana (original) (raw)
Papers by M. Tkachenko
Fundamenta Mathematicae, 2002
Topologies τ 1 and τ 2 on a set X are called T 1-complementary if τ 1 ∩ τ 2 = {X \ F : F ⊆ X is f... more Topologies τ 1 and τ 2 on a set X are called T 1-complementary if τ 1 ∩ τ 2 = {X \ F : F ⊆ X is finite} ∪ {∅} and τ 1 ∪ τ 2 is a subbase for the discrete topology on X. Topological spaces (X, τ X) and (Y, τ Y) are called T 1-complementary provided that there exists a bijection f : X → Y such that τ X and {f −1 (U) : U ∈ τ 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).
Topology and its Applications, 2005
Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the ... more Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the least upper bound τ 1 ∨ τ 2 of τ 1 and τ 2 is the discrete topology. We show that an infinite totally bounded topological group never admits a transversal group topology and we obtain a new criterion for precompactness in lattice-theoretical terms (the existence of transversal group topologies) for a large class of Abelian groups containing all divisible and all finitely generated groups. A full description is given of the class M of all abstract Abelian groups G where this criterion is valid (i.e., when the non-precompact group topologies on G are precisely the transversable ones). It turns out that M is the class of groups G for which the submaximal topology is precompact. We also give a complete description of the structure of the transversable locally compact Abelian groups.
Proceedings of the American Mathematical Society
We prove under the assumption of Martin's Axiom that every precompact Abelian group of size l... more We prove under the assumption of Martin's Axiom that every precompact Abelian group of size less than or equal to 2^\omega belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.
Topology and its Applications, 2006
Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the ... more Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the least upper bound τ 1 ∨ τ 2 of τ 1 and τ 2 is the discrete topology. We give a complete description of the transversable locally compact groups in the case they are connected (earlier, the authors gave such a description in the abelian case). In particular, a connected Lie group is transversable if and only if its center is not compact.
Journal of Pure and Applied Algebra, 2012
Journal of Pure and Applied Algebra, 2000
Forum Mathematicum, 2003
We prove under the assumption of Martin's Axiom that every precompact Abelian group of size ≤ 2 ℵ... more We prove under the assumption of Martin's Axiom that every precompact Abelian group of size ≤ 2 ℵ 0 belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.
According to the celebrated theorem of Comfort and Ross (1966), the product of an arbitrary famil... more According to the celebrated theorem of Comfort and Ross (1966), the product of an arbitrary family of pseudocompact topological groups is pseudocompact. We present an overview of several important generalizations of this result, both of “absolute” and “relative” nature. One of them is the preservation of functional boundedness for subsets of topological groups. Also we consider close notions of C-compactness and r-pseudocompactness for subsets of Tychonoff spaces and establish their productivity in the class of topological groups. Finally, we give a very brief overview of productivity properties in paratopological and semitopological groups.
Pseudocompact Topological Spaces
Topological groups constitute a very special subclass of topological spaces. Every topological gr... more Topological groups constitute a very special subclass of topological spaces. Every topological group satisfying the \(T_0\) separation axiom is automatically Tychonoff, which means that in the class of topological groups, the axioms of separation \(T_0\), \(T_1\), \(T_2\), \(T_3\) and \(T_{3.5}\) are all equivalent.
Topology and its Applications
Abstract Let X = ∏ i ∈ I X i be a product of non-compact spaces. We show that if | I | > ω , t... more Abstract Let X = ∏ i ∈ I X i be a product of non-compact spaces. We show that if | I | > ω , then the remainder Y = b X ∖ X is pseudocompact, for any compactification bX of X. In fact, this theorem follows from a more general result about spaces with an ω-directed lattice of d-open mappings. Under the additional assumption that the space X has countable cellularity, we prove that the remainder Y is C-embedded in bX and that β Y = b X . We apply these results to the remainders of topological groups and spaces of continuous functions with the pointwise convergence topology. For example, we prove that if X is an uncountable space and G is a non-compact topological group, then every remainder of C p ( X , G ) is pseudocompact provided C p ( X , G ) is dense in G X .
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Topology and its Applications
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Tsukuba Journal of Mathematics
We prove that, if X is a Tychonoff connected space and X(x, X) ::;; OJ for some x E X, then there... more We prove that, if X is a Tychonoff connected space and X(x, X) ::;; OJ for some x E X, then there exists a strictly stronger Tychonoff connected topology on the space X, i.e., the space X is not maximal Tychonoff connected. We also establish that if X is locally connected or CT-compact or has pointwise countable type then X cannot be maximal Tychonoff connected.
Applied General Topology
The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychon... more The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychonoff topology is considered. We show that every Tychonoff space X satisfying ω (X) ≤ c and c (X) ≤ N0 admits a finer strongly σ-discrete connected Tychonoff topology of weight 2c. We also prove that every connected Tychonoff space is an open continuous image of a connected strongly σ-discrete submetrizable Tychonoff space. The latter result is applied to represent every connected topological group as a quotient of a connected strongly σ-discrete submetrizable topological group.
Israel Journal of Mathematics
The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infi... more The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.
Topology and its Applications
To the memory of Wistar Comfort (1933-2016), a great topologist and man, to whom we owe much of o... more To the memory of Wistar Comfort (1933-2016), a great topologist and man, to whom we owe much of our inspiration Abstract. We prove that if H is a topological group such that all closed subgroups of H are separable, then the product G × H has the same property for every separable compact group G. Let c be the cardinality of the continuum. Assuming 2 ω 1 = c, we show that there exist: • pseudocompact topological abelian groups G and H such that all closed subgroups of G and H are separable, but the product G × H contains a closed non-separable σ-compact subgroup; • pseudocomplete locally convex vector spaces K and L such that all closed vector subspaces of K and L are separable, but the product K × L contains a closed non-separable σ-compact vector subspace.
Mathematical Notes of the Academy of Sciences of the USSR
ABSTRACT
Siberian Mathematical Journal
Fundamenta Mathematicae, 2002
Topologies τ 1 and τ 2 on a set X are called T 1-complementary if τ 1 ∩ τ 2 = {X \ F : F ⊆ X is f... more Topologies τ 1 and τ 2 on a set X are called T 1-complementary if τ 1 ∩ τ 2 = {X \ F : F ⊆ X is finite} ∪ {∅} and τ 1 ∪ τ 2 is a subbase for the discrete topology on X. Topological spaces (X, τ X) and (Y, τ Y) are called T 1-complementary provided that there exists a bijection f : X → Y such that τ X and {f −1 (U) : U ∈ τ 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).
Topology and its Applications, 2005
Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the ... more Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the least upper bound τ 1 ∨ τ 2 of τ 1 and τ 2 is the discrete topology. We show that an infinite totally bounded topological group never admits a transversal group topology and we obtain a new criterion for precompactness in lattice-theoretical terms (the existence of transversal group topologies) for a large class of Abelian groups containing all divisible and all finitely generated groups. A full description is given of the class M of all abstract Abelian groups G where this criterion is valid (i.e., when the non-precompact group topologies on G are precisely the transversable ones). It turns out that M is the class of groups G for which the submaximal topology is precompact. We also give a complete description of the structure of the transversable locally compact Abelian groups.
Proceedings of the American Mathematical Society
We prove under the assumption of Martin's Axiom that every precompact Abelian group of size l... more We prove under the assumption of Martin's Axiom that every precompact Abelian group of size less than or equal to 2^\omega belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.
Topology and its Applications, 2006
Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the ... more Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the least upper bound τ 1 ∨ τ 2 of τ 1 and τ 2 is the discrete topology. We give a complete description of the transversable locally compact groups in the case they are connected (earlier, the authors gave such a description in the abelian case). In particular, a connected Lie group is transversable if and only if its center is not compact.
Journal of Pure and Applied Algebra, 2012
Journal of Pure and Applied Algebra, 2000
Forum Mathematicum, 2003
We prove under the assumption of Martin's Axiom that every precompact Abelian group of size ≤ 2 ℵ... more We prove under the assumption of Martin's Axiom that every precompact Abelian group of size ≤ 2 ℵ 0 belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.
According to the celebrated theorem of Comfort and Ross (1966), the product of an arbitrary famil... more According to the celebrated theorem of Comfort and Ross (1966), the product of an arbitrary family of pseudocompact topological groups is pseudocompact. We present an overview of several important generalizations of this result, both of “absolute” and “relative” nature. One of them is the preservation of functional boundedness for subsets of topological groups. Also we consider close notions of C-compactness and r-pseudocompactness for subsets of Tychonoff spaces and establish their productivity in the class of topological groups. Finally, we give a very brief overview of productivity properties in paratopological and semitopological groups.
Pseudocompact Topological Spaces
Topological groups constitute a very special subclass of topological spaces. Every topological gr... more Topological groups constitute a very special subclass of topological spaces. Every topological group satisfying the \(T_0\) separation axiom is automatically Tychonoff, which means that in the class of topological groups, the axioms of separation \(T_0\), \(T_1\), \(T_2\), \(T_3\) and \(T_{3.5}\) are all equivalent.
Topology and its Applications
Abstract Let X = ∏ i ∈ I X i be a product of non-compact spaces. We show that if | I | > ω , t... more Abstract Let X = ∏ i ∈ I X i be a product of non-compact spaces. We show that if | I | > ω , then the remainder Y = b X ∖ X is pseudocompact, for any compactification bX of X. In fact, this theorem follows from a more general result about spaces with an ω-directed lattice of d-open mappings. Under the additional assumption that the space X has countable cellularity, we prove that the remainder Y is C-embedded in bX and that β Y = b X . We apply these results to the remainders of topological groups and spaces of continuous functions with the pointwise convergence topology. For example, we prove that if X is an uncountable space and G is a non-compact topological group, then every remainder of C p ( X , G ) is pseudocompact provided C p ( X , G ) is dense in G X .
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Topology and its Applications
Czechoslovak Mathematical Journal
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Tsukuba Journal of Mathematics
We prove that, if X is a Tychonoff connected space and X(x, X) ::;; OJ for some x E X, then there... more We prove that, if X is a Tychonoff connected space and X(x, X) ::;; OJ for some x E X, then there exists a strictly stronger Tychonoff connected topology on the space X, i.e., the space X is not maximal Tychonoff connected. We also establish that if X is locally connected or CT-compact or has pointwise countable type then X cannot be maximal Tychonoff connected.
Applied General Topology
The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychon... more The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychonoff topology is considered. We show that every Tychonoff space X satisfying ω (X) ≤ c and c (X) ≤ N0 admits a finer strongly σ-discrete connected Tychonoff topology of weight 2c. We also prove that every connected Tychonoff space is an open continuous image of a connected strongly σ-discrete submetrizable Tychonoff space. The latter result is applied to represent every connected topological group as a quotient of a connected strongly σ-discrete submetrizable topological group.
Israel Journal of Mathematics
The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infi... more The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.
Topology and its Applications
To the memory of Wistar Comfort (1933-2016), a great topologist and man, to whom we owe much of o... more To the memory of Wistar Comfort (1933-2016), a great topologist and man, to whom we owe much of our inspiration Abstract. We prove that if H is a topological group such that all closed subgroups of H are separable, then the product G × H has the same property for every separable compact group G. Let c be the cardinality of the continuum. Assuming 2 ω 1 = c, we show that there exist: • pseudocompact topological abelian groups G and H such that all closed subgroups of G and H are separable, but the product G × H contains a closed non-separable σ-compact subgroup; • pseudocomplete locally convex vector spaces K and L such that all closed vector subspaces of K and L are separable, but the product K × L contains a closed non-separable σ-compact vector subspace.
Mathematical Notes of the Academy of Sciences of the USSR
ABSTRACT
Siberian Mathematical Journal