A generalization of Čech-complete spaces and Lindelöf Σ -spaces

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 2, page 121-139
  • ISSN: 0010-2628

Abstract

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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 ≤ 2 ω , 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 Σ -spaces: X is an s -space if and only if some (every) remainder of X in a compactification is a Lindelöf Σ -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.

How to cite

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Arhangel'skii, Aleksander V.. "A generalization of Čech-complete spaces and Lindelöf $\Sigma $-spaces." Commentationes Mathematicae Universitatis Carolinae 54.2 (2013): 121-139. <http://eudml.org/doc/252528>.

@article{Arhangelskii2013,
abstract = {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 $\le 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.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {metrizable; Lindelöf $p$-space; Lindelöf $\Sigma $-space; remainder; compactification; $\sigma $-space; countable network; countable type; perfect mapping; Lindelöf -space; Lindelöf -space; remainder; compactification; -space; countable network; perfect mapping},
language = {eng},
number = {2},
pages = {121-139},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A generalization of Čech-complete spaces and Lindelöf $\Sigma $-spaces},
url = {http://eudml.org/doc/252528},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - A generalization of Čech-complete spaces and Lindelöf $\Sigma $-spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 2
SP - 121
EP - 139
AB - 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 $\le 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.
LA - eng
KW - metrizable; Lindelöf $p$-space; Lindelöf $\Sigma $-space; remainder; compactification; $\sigma $-space; countable network; countable type; perfect mapping; Lindelöf -space; Lindelöf -space; remainder; compactification; -space; countable network; perfect mapping
UR - http://eudml.org/doc/252528
ER -

References

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