- Title
- Separability of topological groups : A survey with open problems
- Creator
- Leiderman, Arkady; Morris, Sidney
- Date
- 2018
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/167978
- Identifier
- vital:13787
- Identifier
-
https://doi.org/10.3390/axioms8010003
- Identifier
- ISBN:2075-1680
- 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 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.
- Publisher
- MDPI AG
- Relation
- Axioms Vol. 8, no. 1 (2018), p. 1-18
- Rights
- http://creativecommons.org/licenses/by/4.0/
- Rights
- Copyright © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
- Rights
- Open Access
- Rights
- This metadata is freely available under a CCO license
- Subject
- Free topological group; Isomorphic embedding; Product; Quotient group; Separable topological group; Subgroup
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