Products of topological groups in which all closed subgroups are separable

Arkady G. Leiderman; Mikhail G. Tkachenko

doi:10.1016/j.topol.2018.03.022

Products of topological groups in which all closed subgroups are separable

Arkady G. Leiderman, Mikhail G. Tkachenko

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

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^ω₁=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.

Original language	American English
Pages (from-to)	89-101
Number of pages	13
Journal	Topology and its Applications
Volume	241
DOIs	https://doi.org/10.1016/j.topol.2018.03.022
State	Published - 1 Jun 2018

Keywords

Closed subgroup
Locally convex space
Pseudocompact
Pseudocomplete
Separable
Topological group

All Science Journal Classification (ASJC) codes

Geometry and Topology

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10.1016/j.topol.2018.03.022

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title = "Products of topological groups in which all closed subgroups are separable",

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 2ω1 =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.",

keywords = "Closed subgroup, Locally convex space, Pseudocompact, Pseudocomplete, Separable, Topological group",

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note = "Publisher Copyright: {\textcopyright} 2018 Elsevier B.V.",

year = "2018",

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doi = "10.1016/j.topol.2018.03.022",

language = "American English",

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T1 - Products of topological groups in which all closed subgroups are separable

AU - Leiderman, Arkady G.

AU - Tkachenko, Mikhail G.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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 2ω1 =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.

AB - 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 2ω1 =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.

KW - Closed subgroup

KW - Locally convex space

KW - Pseudocompact

KW - Pseudocomplete

KW - Separable

KW - Topological group

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U2 - 10.1016/j.topol.2018.03.022

DO - 10.1016/j.topol.2018.03.022

M3 - Article

SN - 0166-8641

VL - 241

SP - 89

EP - 101

JO - Topology and its Applications

JF - Topology and its Applications

ER -

Products of topological groups in which all closed subgroups are separable

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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