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Title: On subcompactness and countable subcompactness of metrizable spaces in ZF (English)
Author: Keremedis, Kyriakos
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 63
Issue: 2
Year: 2022
Pages: 229-244
Summary lang: English
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Category: math
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Summary: We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space $\mathbf{X}=(X,T)$ is countably compact if and only if it is countably subcompact relative to $T$. (iii) For every metrizable space $\mathbf{X}=(X,T)$, the following are equivalent: \noindent(a) $\mathbf{X}$ is compact; \noindent(b) for every open filter $\mathcal{F}$ of $\mathbf{X}$, $\bigcap \{\overline{F}\colon F\in \mathcal{F}\}\neq \emptyset $; \noindent(c) $\mathbf{X}$ is subcompact relative to $T$. We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable space is completely metrizable, (b) every countably subcompact metrizable space is subcompact, (c) every completely metrizable space is subcompact, is relatively consistent with ZF. (v) AC if and only if for every family $\{\mathbf{X}_{i}\colon i\in I\}$ of metrizable subcompact spaces, for every family $\{\mathcal{B}_{i}\colon i\in I\}$ such that for every $i\in I$, $\mathcal{B}_{i}$ is a subcompact base for $\mathbf{X}_{i}$, the Tychonoff product $\mathbf{X}=\prod_{i\in I} \mathbf{X}_{i}$ is subcompact with respect to the standard base $\mathcal{B}$ of $\mathbf{X}$ generated by the family $\{\mathcal{B}_{i}\colon i\in I\}$. (English)
Keyword: axiom of choice
Keyword: compact
Keyword: countably compact
Keyword: subcompact
Keyword: countably subcompact
Keyword: lightly compact metric space
MSC: 03E25
MSC: 54D30
MSC: 54E35
MSC: 54E45
MSC: 54E50
idZBL: Zbl 07613032
idMR: MR4506134
DOI: 10.14712/1213-7243.2022.018
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Date available: 2022-11-02T09:19:59Z
Last updated: 2024-07-01
Stable URL: http://hdl.handle.net/10338.dmlcz/151087
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