Some versions of second countability of metric spaces in ZF and their role to compactness

Kyriakos Keremedis

Commentationes Mathematicae Universitatis Carolinae (2018)

  • Volume: 59, Issue: 1, page 119-134
  • ISSN: 0010-2628

Abstract

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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 ε > 0 , every cover by open balls of radius ε 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 ℝ holds true. (iii) A countably compact metric space is separable if and only if it is second countable.

How to cite

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Keremedis, Kyriakos. "Some versions of second countability of metric spaces in ZF and their role to compactness." Commentationes Mathematicae Universitatis Carolinae 59.1 (2018): 119-134. <http://eudml.org/doc/294495>.

@article{Keremedis2018,
abstract = {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.},
author = {Keremedis, Kyriakos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {axiom of choice; compact space; countably compact space; totally bounded space; Lindelöf space; separable space; second countable metric space},
language = {eng},
number = {1},
pages = {119-134},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some versions of second countability of metric spaces in ZF and their role to compactness},
url = {http://eudml.org/doc/294495},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Keremedis, Kyriakos
TI - Some versions of second countability of metric spaces in ZF and their role to compactness
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 1
SP - 119
EP - 134
AB - 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.
LA - eng
KW - axiom of choice; compact space; countably compact space; totally bounded space; Lindelöf space; separable space; second countable metric space
UR - http://eudml.org/doc/294495
ER -

References

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