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.


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


Zbl 0543.00009