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

top
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

top

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

top
  1. Bentley H. L., Herrlich H., 10.1016/S0166-8641(97)00138-7, Topology Appl. 85 (1998), 153–164. Zbl0922.03068DOI10.1016/S0166-8641(97)00138-7
  2. Brunner N., Lindelöf Räume und Auswahlaxiom, Anz. Österr. Akad. Wiss. Math.-Nat. 119 (1982), 161–165. 
  3. Good C., Tree I. J., Watson S., 10.1090/S0002-9939-98-04163-X, Proc. Amer. Math. Soc. 126 (1998), 1211–1218. DOI10.1090/S0002-9939-98-04163-X
  4. Herrlich H., Axiom of Choice, Lecture Notes in Mathematics, 1876, Springer, Berlin, 2006. Zbl1102.03049
  5. 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. 
  6. Herrlich H., Strecker G. E., When is ℕ Lindelöf?, Comment. Math. Univ. Carolin. 38 (1997), no. 3, 553–556. 
  7. Howard P., Keremedis K., Rubin J. E., Stanley A., 10.1002/(SICI)1521-3870(200005)46:2<219::AID-MALQ219>3.0.CO;2-2, Math. Log. Q. 46 (2000), no. 2, 219–232. DOI10.1002/(SICI)1521-3870(200005)46:2<219::AID-MALQ219>3.0.CO;2-2
  8. Howard P., Rubin J. E., 10.1090/surv/059, Math. Surveys and Monographs, 59, American Mathematical Society, Providence, 1998. Zbl0947.03001DOI10.1090/surv/059
  9. Keremedis K., 10.1016/j.topol.2012.08.003, Topology Appl. 159 (2012), 3396–3403. DOI10.1016/j.topol.2012.08.003
  10. Keremedis K., 10.1016/j.topol.2016.07.021, Topology Appl. 210 (2016), 366–375. DOI10.1016/j.topol.2016.07.021
  11. Keremedis K., 10.4064/ba8087-12-2016, Bull. Polish Acad. Sci. Math. 64 (2016), 109–136. DOI10.4064/ba8087-12-2016
  12. Keremedis K., Tachtsis E., 10.1002/1521-3870(200101)47:1<117::AID-MALQ117>3.0.CO;2-N, MLQ Math. Log. Q. 47 (2001), 117–128. DOI10.1002/1521-3870(200101)47:1<117::AID-MALQ117>3.0.CO;2-N
  13. Munkres J. R., Topology, Prentice-Hall, New Jersey, 1975. Zbl0951.54001
  14. Tachtsis E., Disasters in metric topology without choice, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 165–174. 

NotesEmbed ?

top

You must be logged in to post comments.