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

Foged, L.

Topology, Proc. Conf., Houston/Tex. 1983, Topology Proc. 8, No. 2, 259-266 (1983).

[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.

Reviewer: John W. Baker (Sheffield)

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

Citations:

Zbl 0543.00009