Article
| 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: | 2023-03-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151087 |
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| Reference: | [1] Brunner N.: Lindelöf Räume und Auswahlaxiom.Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 119 (1982), no. 9, 161–165 (German, English summary). MR 0728812 |
| Reference: | [2] Brunner N.: Kategoriesätze und multiples Auswahlaxiom.Z. Math. Logik Grundlagen Math. 29 (1983), no. 5, 435–443 (German). MR 0716858, 10.1002/malq.19830290804 |
| Reference: | [3] Dow A., Porter J. R., Stephenson R. M., Jr., Woods R. G.: Spaces whose pseudocompact subspaces are closed subsets.Appl. Gen. Topol. 5 (2004), no. 2, 243–264. Zbl 1066.54024, MR 2121792, 10.4995/agt.2004.1973 |
| Reference: | [4] Good C., Tree I. J.: Continuing horrors of topology without choice.Topology Appl. 63 (1995), no. 1, 79–90. MR 1328621, 10.1016/0166-8641(95)90010-1 |
| Reference: | [5] de Groot J.: Subcompactness and the Baire category theorem.Nederl. Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 761–767. MR 0159303, 10.1016/S1385-7258(63)50076-6 |
| Reference: | [6] Herrlich H.: Products of Lindelöf $T_{2}$-spaces are Lindelöf – in some models of ZF.Comment. Math. Univ. Carolin. 43 (2002), no. 2, 319–333. MR 1922130 |
| Reference: | [7] Howard P., Rubin J. E.: Consequences of the Axiom of Choice.Mathematical Surveys and Monographs, 59, American Mathematical Society, Providence, 1998. Zbl 0947.03001, MR 1637107, 10.1090/surv/059 |
| Reference: | [8] Ikeda Y.: Čech-completeness and countably subcompactness.Topology Proc. 14 (1989), no. 1, 75–87. MR 1081121 |
| Reference: | [9] Keremedis K.: On the relative strength of forms of compactness of metric spaces and their countable productivity in ZF.Topology Appl. 159 (2012), no. 16, 3396–3403. MR 2964853, 10.1016/j.topol.2012.08.003 |
| Reference: | [10] Keremedis K.: On sequential compactness and related notions of compactness of metric spaces in ZF.Bull. Pol. Acad. Sci. Math. 64 (2016), no. 1, 29–46. MR 3550611, 10.4064/ba8087-12-2016 |
| Reference: | [11] Keremedis K.: On pseudocompactness and light compactness of metric spaces in ZF.Bull. Pol. Acad. Sci. Math. 66 (2018), no. 2, 99–113. MR 3892754, 10.4064/ba8131-10-2018 |
| Reference: | [12] Keremedis K.: On lightly and countably compact spaces in ZF.Quaest. Math. 42 (2019), no. 5, 579–592. MR 3969540, 10.2989/16073606.2018.1463300 |
| Reference: | [13] Keremedis K., Wajch E.: On Loeb and sequential spaces in ZF.Topology Appl. 280 (2020), 107279, 21 pages. MR 4108491, 10.1016/j.topol.2020.107279 |
| Reference: | [14] Mardešić S., Papić P.: Sur les espaces dont toute transformation réelle continue est bornée.Hrvatsko Prirod. Društvo. Glasnik Mat.-Fiz. Astr. Ser. II. 10 (1955), 225–232 (French, Serbo-Croatian summary). MR 0080292 |
| Reference: | [15] Stone A. H.: Hereditarily compact spaces.Amer. J. Math. 82 (1960), 900–914. MR 0120620, 10.2307/2372948 |
| Reference: | [16] Willard S.: General Topology.Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, 1970. Zbl 1052.54001, MR 0264581 |
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