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Separability of Topological Groups: A Survey with Open Problems

Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel
Centre for Informatics and Applied Optimization (CIAO), Federation University Australia, P.O. Box 663, Ballarat 3353, Australia
Department of Mathematics and Statistics, La Trobe University, Melbourne 3086, Australia
These authors contributed equally to this work.
Author to whom correspondence should be addressed.
Received: 25 November 2018 / Revised: 23 December 2018 / Accepted: 25 December 2018 / Published: 29 December 2018
(This article belongs to the Collection Topological Groups)
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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 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. View Full-Text
Keywords: separable topological group; subgroup; product; isomorphic embedding; quotient group; free topological group separable topological group; subgroup; product; isomorphic embedding; quotient group; free topological group
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Leiderman, A.G.; Morris, S.A. Separability of Topological Groups: A Survey with Open Problems. Axioms 2019, 8, 3.

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