Separability of topological groups: A survey with open problems

Arkady G. Leiderman, Sidney A. Morris

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

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 ℓp, 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.

Original languageEnglish
Article number3
JournalAxioms
Volume8
Issue number1
DOIs
StatePublished - 29 Dec 2018

Keywords

  • Free topological group
  • Isomorphic embedding
  • Product
  • Quotient group
  • Separable topological group
  • Subgroup

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Mathematical Physics
  • Logic
  • Geometry and Topology

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