The separable quotient problem for topological groups

Abstract

The famous Banach-Mazur problem, which asks if every infinitedimensional 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 Fréchet 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 σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.Problem 1.1. Separable Quotient Problem for Banach Spaces. Does every infinite-dimensional Banach space have a quotient Banach space which is separable and infinite-dimensional? The related Quotient Schauder Basis Problem for Banach Spaces is due to Aleksander (Olek) Pe lczyński [24]. Problem 1.2. Quotient Schauder Basis Problem for Banach Spaces. Does every infinite-dimensional Banach space have a quotient Banach space which is infinite-dimensional and has a Schauder basis?

A topological group observation on the Banach–Mazur separable quotient problem

“…Remark 2.6. We mention that A. Leiderman, M. Tkachenko and the second author in their paper [12] have examined the question: which topological groups G have separable quotient groups? Their results include substantial information on the cases that G is (i) a compact group, (ii) a pro-Lie group, (iii) a pseudocompact group and (iv) a precompact group.…”

The Tubby Torus as a Quotient Group

“…It follows from Theorem 2.5 of [3] that every non-metrizable connected locally compact abelian group has the tubby torus as a quotient group. But as a connected locally compact abelian group G is isomorphic as a topological group to the product R n × K, for some non-negative integer n and compact abelian group K, and R n and all compact metrizable groups are separable, we see that if G is non-separable then it is non-metrizable.…”

“…The following problem stated in [3] is also unsolved, but a negative answer to it would give a negative answer to Problem 1. Problem 2.…”

Separability of Topological Groups: A Survey with Open Problems

“…The main aim of this survey paper is to present systematically the results concerning the behavior of separability of topological groups with respect to the topological operations listed above and make clear which problems are open. Much of the material is from the recent publications [1][2][3][4][5][6][7][8].…”

“…Now we formulate various natural versions of the Separable Quotient Problem(s) for Topological Groups. Unless explicitly stated otherwise the results presented in this section are from the paper[5]. Does every non-totally disconnected topological group have a quotient group which is a non-trivial separable topological group?…”