×

The Baire category theorem for Fréchet groups in which every null sequence has a summable subsequence. (English) Zbl 0557.22002

[For the entire collection see Zbl 0543.00009.]
A topological group has property (K) if every sequence \((x_ n)\) which converges to 0 has a subsequence \((x_{n_ k})\) for which \(\sum x_{n_ k}\) is a convergent series. Clearly every complete metrisable topological group has property (K). A topological space is sequential if every sequentially closed set is closed and is Fréchet if the sequential closure of any set is equal to its closure. It is shown that a Hausdorff Fréchet topological group with property (K) is a Baire space. This result is shown to be false if we only assume sequential rather than Fréchet.

MSC:

22A05 Structure of general topological groups
46A35 Summability and bases in topological vector spaces
54H99 Connections of general topology with other structures, applications

Citations:

Zbl 0543.00009
PDF BibTeX XML Cite