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Keywords:
metrizable; Lindelöf $p$-space; Lindelöf $\Sigma $-space; remainder; compactification; $\sigma $-space; countable network; countable type; perfect mapping
metrizable; Lindelöf $p$-space; Lindelöf $\Sigma $-space; remainder; compactification; $\sigma $-space; countable network; countable type; perfect mapping
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 [Arhangel'skii A.V., Remainders of metrizable and close to metrizable spaces, Fund. Math. {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 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.
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