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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.

##### MSC:
 54D40 Remainders in general topology 54B05 Subspaces in general topology 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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