The separable quotient problem for topological groups
The separable quotient problem for topological groups
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DOI：
10.1007/s11856-019-1931-1
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发表时间：
2017-07
影响因子：
1
通讯作者：
A. Leiderman;S. Morris;M. Tkachenko
中科院分区：
文献类型：
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作者：
A. Leiderman;S. Morris;M. Tkachenko
著名的巴拿赫-马祖尔问题，它问是否每个无限维的Banach空间有一个无限维的可分商Banach空间，一直没有解决85年，虽然它已回答肯定的自反Banach空间，甚至Banach空间是可分的。局部凸空间的类似问题已经得到了否定的回答，但是已经证明对于包括所有非赋范Fréchet空间在内的大类局部凸空间都是成立的。对于拓扑群G，存在四个自然的类似问题：G是否有一个可分商群（i）非平凡的;（ii）无限的;（iii）可度量化的;（iv）无限可度量化的。对于属于以下重要类的群G，证明了所有四个问题的正答案：（a）所有紧群;（B）所有局部紧交换群;（c）所有σ-紧局部紧群;（d）所有交换pro-Lie群;（e）所有σ-紧pro-Lie群;（f）所有伪紧群。然而，一个令人惊讶的例子，一个不可数的预紧群G的生产，没有非平凡的可分商群以外的平凡群。IndeedGτ具有相同的性质，对于每个基数τ≥ 1。
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 Fréchet spaces.For a topological groupGthere are four natural analogous problems: DoesGhave a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. Positive answers to all four questions are proved for groupsGwhich belong to the important classes of (a) all compact groups; (b) all locally compact abelian groups; (c) allσ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) allσ-compact pro-Lie groups; (f) all pseudocompact groups.However, a surprising example of an uncountable precompact groupGis produced which has no non-trivial separable quotient group other than the trivial group. IndeedGτhas the same property, for every cardinal numberτ≥ 1.