zbMATH — the first resource for mathematics

A generalization of Čech-complete spaces and Lindelöf \(\Sigma \)-spaces. (English) Zbl 1289.54085
Summary: The class of \(s\)-spaces is studied in detail. It includes, in particular, all Čech-complete spaces, Lindelöf \(p\)-spaces, metrizable spaces with the weight \(\leq 2^{\omega }\), but countable non-metrizable spaces and some metrizable spaces are not in it. It is shown that \(s\)-spaces are in a duality with Lindelöf \(\Sigma \)-spaces: \(X\) is an \(s\)-space if and only if some (every) remainder of \(X\) in a compactification is a Lindelöf \(\Sigma \)-space [A. V. Arhangel’skii, Fundam. Math. 220, No. 1, 71–81 (2013; Zbl 1267.54024)]. A basic fact is established: the weight and the networkweight coincide for all \(s\)-spaces. This theorem generalizes the similar statement about Čech-complete spaces. We also study hereditarily \(s\)-spaces, provide various sufficient conditions for a space to be a hereditarily \(s\)-space, and establish that every metrizable space has a dense subspace which is a hereditarily \(s\)-space. It is also shown that every dense-in-itself compact hereditarily \(s\)-space is metrizable.

54D40 Remainders in general topology
54B05 Subspaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
PDF BibTeX Cite
Full Text: EMIS