## A. V. Arhangel'skii

*A generalization of \v Cech-complete spaces and Lindel\"of $\Sigma $-spaces*

Comment.Math.Univ.Carolin. 54,2 (2013) 121-139.**Abstract:**The class of $s$-spaces is studied in detail. It includes, in particular, all \v Cech-complete spaces, Lindel\"of $p$-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 Lindel\"of $\Sigma $-spaces: $X$ is an $s$-space if and only if some (every) remainder of $X$ in a compactification is a Lindel\"of $\Sigma $-space [Arhangel'skii A.V., {\it Remainders of metrizable and close to metrizable spaces\/}, Fund.\ Math.\ {\bf 220} (2013), 71--81]. A~basic fact is established: the weight and the networkweight coincide for all $s$-spaces. This theorem generalizes the similar statement about \v Cech-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.

**Keywords:** metrizable, Lindel\"of $p$-space, Lindel\"of $\Sigma $-space, remainder, compactification, $\sigma $-space, countable network, countable type, perfect mapping

**AMS Subject Classification:** 54A25 54B05

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