2022/11/13 by Dekui Peng, Peng, Dekui
Mathematics · #22A05 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2211.06831
openalex publication_date 2022/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The topological group version of the celebrated Banach-Mazur problem asks wether every infinite topological group has a non-trivial separable quotient group. It is known that compact groups have infinite separable metrizable quotient groups. However, as dense subgroups of compact groups, precompact groups may admit no non-trivial metrizable quotient groups, so also no non-trivial separable quotient groups. In this paper, we study the least cardinal \mathfrakm (resp. \mathfrakn) such that every infinite precompact abelian group admits a quotient group with density character ≤ \mathfrakm (resp. with weight ≤ \mathfrakn). It is shown that if 2^<\mathfrakc=\mathfrakc, then \mathfrakm=\mathfrakc and \mathfrakn=2^\mathfrakc. A more general problem is to describe the set QW(G) of all possible weights of infinite proper quotient groups of a precompact abelian group G. We prove that for every subset E of the interval [ω, \mathfrakc], there exists a precompact abelian group G with QW(G)=E. If ω∈ E, then G can be chosen to be pseudocompact. In an appendix, we give an example to show that a non-totally disconnected locally compact group may admit no separable quotient groups. This answers an open problem posed in \citeLMT.