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- W4309131144 abstract "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 $mathfrak{m}$ (resp. $mathfrak{n}$) such that every infinite precompact abelian group admits a quotient group with density character $leq mathfrak{m}$ (resp. with weight $leq mathfrak{n}$). It is shown that if $2^{<mathfrak{c}}=mathfrak{c}$, then $mathfrak{m}=mathfrak{c}$ and $mathfrak{n}=2^mathfrak{c}$. 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 $[omega, mathfrak{c}]$, there exists a precompact abelian group $G$ with $QW(G)=E$. If $omegain E$, 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 cite{LMT}." @default.
- W4309131144 created "2022-11-23" @default.
- W4309131144 creator A5013197192 @default.
- W4309131144 date "2022-11-13" @default.
- W4309131144 modified "2023-09-24" @default.
- W4309131144 title "Densities and Weights of Quotients of Precompact Abelian Groups" @default.
- W4309131144 doi "https://doi.org/10.48550/arxiv.2211.06831" @default.
- W4309131144 hasPublicationYear "2022" @default.
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