Burzyk, J.; Kliś, C.; Lipecki, Z. On metrizable Abelian groups with a completeness-type property. (English) Zbl 0552.46001 Colloq. Math. 49, 33-39 (1984). We say that an Abelian topological group X has property (K) if every sequence \((x_ n)\) in X with \(x_ n\to 0\) contains a subsequence \((x_{n_ k})\) such that the series \(\sum^{\infty}_{k=1}x_{n_ k}\) is convergent. This property was first isolated by S. Mazur and W. Orlicz [Stud. Math. 13, 137-179 (1953; Zbl 0052.111)], who realized that it can be used as a substitute for completeness in some basic theorems of functional analysis. The present paper yields the following explanation for this: Every metrizable group with property (K) is a Baire space [see L. Foged, Topology, Proc. Conf., Vol. 8, 259- 266 (1983) for a nonmetrizable generalization]. The reverse implication is shown to fail in some general sense. We also construct, by transfinite induction, for a nondiscrete metrizable complete Abelian group of cardinality \(2^{\aleph_ 0}\) satisfying some algebraic assumption (which holds, e.g., in \({\mathbb{R}}\) or \({\mathbb{T}})\), a dense proper subgroup with property (K). Cited in 2 ReviewsCited in 6 Documents MSC: 46A04 Locally convex Fréchet spaces and (DF)-spaces 54E50 Complete metric spaces 54E52 Baire category, Baire spaces 22A10 Analysis on general topological groups Keywords:Abelian topological group; substitute for completeness; Every metrizable group with property (K) is a Baire space; transfinite induction Citations:Zbl 0552.46002; Zbl 0052.111 PDFBibTeX XMLCite \textit{J. Burzyk} et al., Colloq. Math. 49, 33--39 (1984; Zbl 0552.46001) Full Text: DOI