| Title:
|
Some versions of second countability of metric spaces in ZF and their role to compactness (English) |
| Author:
|
Keremedis, Kyriakos |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
1 |
| Year:
|
2018 |
| Pages:
|
119-134 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every $\varepsilon > 0$, every cover by open balls of radius $\varepsilon $ has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of $\mathbb{R}$ holds true. (iii) A countably compact metric space is separable if and only if it is second countable. (English) |
| Keyword:
|
axiom of choice |
| Keyword:
|
compact space |
| Keyword:
|
countably compact space |
| Keyword:
|
totally bounded space |
| Keyword:
|
Lindelöf space |
| Keyword:
|
separable space |
| Keyword:
|
second countable metric space |
| MSC:
|
54E35 |
| MSC:
|
54E45 |
| idZBL:
|
Zbl 06890400 |
| idMR:
|
MR3783812 |
| DOI:
|
10.14712/1213-7243.2015.229 |
| . |
| Date available:
|
2018-04-17T13:52:10Z |
| Last updated:
|
2020-04-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147182 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[7] Howard P., Keremedis K., Rubin J. E., Stanley A.: Paracompactness of metric spaces and the axiom of multiple choice.Math. Log. Q. 46 (2000), no. 2, 219–232. 10.1002/(SICI)1521-3870(200005)46:2<219::AID-MALQ219>3.0.CO;2-2 |
| Reference:
|
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| Reference:
|
[9] Keremedis K.: On the relative strength of forms of compactness of metric spaces and their countable productivity in $\mathbf {ZF}$.Topology Appl. 159 (2012), 3396–3403. 10.1016/j.topol.2012.08.003 |
| Reference:
|
[10] Keremedis K.: On metric spaces where continuous real valued functions are uniformly continuous in $\mathbf {ZF}$.Topology Appl. 210 (2016), 366–375. 10.1016/j.topol.2016.07.021 |
| Reference:
|
[11] Keremedis K.: Some notions of separability of metric spaces in $\mathbf {ZF}$ and their relation to compactness.Bull. Polish Acad. Sci. Math. 64 (2016), 109–136. 10.4064/ba8087-12-2016 |
| Reference:
|
[12] Keremedis K., Tachtsis E.: Compact metric spaces and weak forms of the axiom of choice.MLQ Math. Log. Q. 47 (2001), 117–128. 10.1002/1521-3870(200101)47:1<117::AID-MALQ117>3.0.CO;2-N |
| Reference:
|
[13] Munkres J. R.: Topology.Prentice-Hall, New Jersey, 1975. Zbl 0951.54001 |
| Reference:
|
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| . |
