The topological group version of the celebrated Banach–Mazur problem asks whether 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 separable quotient groups, so also no non-trivial metrizable quotient groups. In this paper, we study the least cardinal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\frak{m}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">m</mi> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\frak{n}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">n</mi> </mrow> </math></EquationSource> </InlineEquation>) such that every infinite precompact abelian group admits a quotient group with density ≤ <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frak{m}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">m</mi> </mrow> </math></EquationSource> </InlineEquation> (resp. with weight ≤ <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frak{n}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">n</mi> </mrow> </math></EquationSource> </InlineEquation>). It is shown that if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2^{<\frak{c}}=\frak{c}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mn>2</mn> <mrow> <mo><</mo> <mrow> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </msup> <mo>=</mo> <mrow> <mi mathvariant="fraktur">c</mi> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\frak{m}=\frak{c}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">m</mi> </mrow> <mo mathvariant="fraktur">=</mo> <mrow> <mi mathvariant="fraktur">c</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\frak{n}=2^{\frak{c}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">n</mi> </mrow> <mo mathvariant="fraktur">=</mo> <msup> <mn mathvariant="fraktur">2</mn> <mrow> <mrow> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </msup> </math></EquationSource> </InlineEquation>.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 <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$[\omega,\frak{c}]$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">[</mo> <mi>ω</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">c</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">]</mo> </math></EquationSource> </InlineEquation>, 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 by Leiderman, Morris and Tkachenko.

Densities and weights of quotients of precompact abelian groups

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发表日期：

2025年9月4日

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Dekui Peng

摘要

The topological group version of the celebrated Banach–Mazur problem asks whether 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 separable quotient groups, so also no non-trivial metrizable quotient groups. In this paper, we study the least cardinal $\frak{m}$ $m$ (resp. $\frak{n}$ $n$ ) such that every infinite precompact abelian group admits a quotient group with density ≤ $\frak{m}$ $m$ (resp. with weight ≤ $\frak{n}$ $n$ ). It is shown that if $2^{<\frak{c}}=\frak{c}$ $2^{< c} = c$ , then $\frak{m}=\frak{c}$ $m = c$ and $\frak{n}=2^{\frak{c}}$ $n = 2^{c}$ .

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 $[\omega,\frak{c}]$ $[ω, c]$ , 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 by Leiderman, Morris and Tkachenko.