## 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

### Keywords:

sequential space; Fréchet topological group; Baire space

Zbl 0543.00009