The Separable Quotient Problem for Topological Groups

TL;DR

This paper investigates the existence of separable quotients in topological groups, providing negative results for general groups and positive results for specific classes like compact and locally compact abelian groups.

Contribution

It introduces four natural analogues of the Banach-Mazur problem for topological groups and establishes their answers, advancing understanding of quotient structures in various classes of topological groups.

Findings

All four questions are answered negatively in general.

Positive results for compact, locally compact abelian, and pro-Lie groups.

Negative results for precompact groups.

Abstract

The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fr\'echet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all $\sigma$-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all $\sigma$-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.