Next Article in Journal
Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays
Previous Article in Journal
Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV
Previous Article in Special Issue
(L)-Semigroup Sums
Article Menu

Export Article

Open AccessArticle

Separability of Topological Groups: A Survey with Open Problems

1
Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel
2
Centre for Informatics and Applied Optimization (CIAO), Federation University Australia, P.O. Box 663, Ballarat 3353, Australia
3
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)
Full-Text   |   PDF [345 KB, uploaded 31 December 2018]   |   Review Reports

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 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).
SciFeed

Share & Cite This Article

MDPI and ACS Style

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

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top