Nonseparable closed vector subspaces of separable topological vector spaces

Ka̧kol, Jerzy; Leiderman, Arkady; Morris, Sidney A.

doi:10.1007/s00605-016-0876-2

Cited by 2 publications

(3 citation statements)

References 15 publications

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Section: Introductionmentioning

confidence: 99%

“…Recently this result was generalized in [9] as follows: If each E i , for i ∈ I, is an lcs with at least c of the E i 's not having the weak topology, then the product i∈I E i contains a closed non-separable vector subspace. These facts prompt the following problem for lcs, similar to the questions considered earlier for topological groups.…”

Section: Introductionmentioning

confidence: 99%

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Products of topological groups in which all closed subgroups are separable

Leiderman

Tkachenko

2018

Topology and its Applications

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Products of topological groups in which all closed subgroups are separable

Leiderman

Tkachenko

2018

Topology and its Applications

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“…The main aim of this survey paper is to present systematically the results concerning the behavior of separability of topological groups with respect to the topological operations listed above and make clear which problems are open. Much of the material is from the recent publications [1][2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Separability of Topological Groups: A Survey with Open Problems

Leiderman

Morris

2018

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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.

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Nonseparable closed vector subspaces of separable topological vector spaces

Cited by 2 publications

References 15 publications

Products of topological groups in which all closed subgroups are separable

Products of topological groups in which all closed subgroups are separable

Separability of Topological Groups: A Survey with Open Problems

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