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40101 Pure Mathematics
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Density character of subgroups of topological groups

- Leiderman, Arkady, Morris, Sidney, Tkachenko, Mikhail

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

Separability of topological groups : A survey with open problems

- Leiderman, Arkady, Morris, Sidney

**Authors:**Leiderman, Arkady , Morris, Sidney**Date:**2018**Type:**Text , Journal article**Relation:**Axioms Vol. 8, no. 1 (2018), p. 1-18**Full Text:****Reviewed:****Description:**Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C(K) for metrizable compact spaces K; and ℓ p , for p ≥ 1. This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey. © 2018 by the authors.

**Authors:**Leiderman, Arkady , Morris, Sidney**Date:**2018**Type:**Text , Journal article**Relation:**Axioms Vol. 8, no. 1 (2018), p. 1-18**Full Text:****Reviewed:****Description:**Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C(K) for metrizable compact spaces K; and ℓ p , for p ≥ 1. This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey. © 2018 by the authors.

Embedding of the free abelian topological group A (X ⊕ X) into A (X)

- Krupski, Mikolaj, Leiderman, Arkady, Morris, Sidney

**Authors:**Krupski, Mikolaj , Leiderman, Arkady , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Mathematika Vol. 65, no. 3 (2019), p. 708-718**Full Text:**false**Reviewed:****Description:**We consider the following question: for which metrizable separable spaces X does the free abelian topological group A (X ⊕ X) isomorphically embed into A (X). While for many natural spaces X such an embedding exists, our main result shows that if X is a Cook continuum or X is a rigid Bernstein set, then A(X ⊕ X) does not embed into A(X) as a topological subgroup. The analogous statement is true for the free boolean group B (X).**Description:**We consider the following question: for which metrizable separable spaces X does the free abelian topological group A (X

Varieties of abelian topological groups with coproducts

- Gabriyelyan, Saak, Leiderman, Arkady, Morris, Sidney

**Authors:**Gabriyelyan, Saak , Leiderman, Arkady , Morris, Sidney**Date:**2015**Type:**Text , Journal article**Relation:**Algebra Universalis Vol. 74, no. 3-4 (2015), p. 241-251**Full Text:**false**Reviewed:****Description:**Varieties of groups, introduced in the 1930s by Garret Birkhoff and B.H. Neumann, are defined as classes of groups satisfying certain laws or equivalently as classes of groups closed under the formation of subgroups, quotient groups, and arbitrary cartesian products. In the 1960s the third author introduced varieties of topological groups as classes of (not necessarily Hausdorff) topological groups closed under subgroups, quotient groups and cartesian products with the Tychonoff topology. While there is only a countable number of varieties of abelian groups, there is a proper class of varieties of abelian topological groups. We observe that while every variety of abelian groups is closed under abelian coproducts, varieties of abelian topological groups are in general not closed under abelian coproducts with the coproduct topology. So this paper studies varieties of abelian topological groups which are also closed under abelian coproducts with the coproduct topology. Noting that the variety of all abelian groups is singly generated, that is, it is the smallest variety containing some particular group, but that the variety of all abelian topological groups is not singly generated, it is proved here that the variety of all abelian topological groups with coproducts is indeed singly generated. There is much literature describing varieties of topological groups generated by various classical topological groups, and the study of varieties with coproducts generated by particular classical topological groups is begun here. Some nice results are obtained about those varieties of abelian topological groups with coproducts which are also closed with regard to forming Pontryagin dual groups. © 2015, Springer Basel.

Nonseparable closed vector subspaces of separable topological vector spaces

- Kakol, Jerzy, Leiderman, Arkady, Morris, Sidney

**Authors:**Kakol, Jerzy , Leiderman, Arkady , Morris, Sidney**Date:**2017**Type:**Text , Journal article**Relation:**Monatshefte Fur Mathematik Vol. 182, no. 1 (2017), p. 39-47**Full Text:**false**Reviewed:****Description:**In 1983 P. Domanski investigated the question: For which separable topological vector spaces E, does the separable space have a nonseparable closed vector subspace, where is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line the space of all continuous real-valued functions on endowed with the pointwise convergence topology, contains a nonseparable closed vector subspace while is separable.**Description:**In 1983 P. DomaA"ski investigated the question: For which separable topological vector spaces E, does the separable space have a nonseparable closed vector subspace, where is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line the space of all continuous real-valued functions on endowed with the pointwise convergence topology, contains a nonseparable closed vector subspace while is separable.

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