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Keywords:
axiom of choice; compact space; countably compact space; totally bounded space; Lindelöf space; separable space; second countable metric space
axiom of choice; compact space; countably compact space; totally bounded space; Lindelöf space; separable space; second countable metric space
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.
References:
[1] Bentley H. L., Herrlich H.: Countable choice and pseudometric spaces. Topology Appl. 85 (1998), 153–164. DOI 10.1016/S0166-8641(97)00138-7 | Zbl 0922.03068
[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.: On Stone's theorem and the axiom of choice. Proc. Amer. Math. Soc. 126 (1998), 1211–1218. DOI 10.1090/S0002-9939-98-04163-X
[4] Herrlich H.: Axiom of Choice. Lecture Notes in Mathematics, 1876, Springer, Berlin, 2006. Zbl 1102.03049
[5] Herrlich H.: Products of Lindelöf $T_{2}$-spaces are Lindelöf---in some models of $ {\rm {ZF}}$. Comment. Math. Univ. Carolin. 43, (2002), no. 2, 319–333.
[6] Herrlich H., Strecker G. E.: When is $\mathbb N$ Lindelöf?. Comment. Math. Univ. Carolin. 38 (1997), no. 3, 553–556.
[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. DOI 10.1002/(SICI)1521-3870(200005)46:2<219::AID-MALQ219>3.0.CO;2-2
[8] Howard P., Rubin J. E.: Consequences of the Axiom of Choice. Math. Surveys and Monographs, 59, American Mathematical Society, Providence, 1998. DOI 10.1090/surv/059 | Zbl 0947.03001
[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. DOI 10.1016/j.topol.2012.08.003
[10] Keremedis K.: On metric spaces where continuous real valued functions are uniformly continuous in $\mathbf {ZF}$. Topology Appl. 210 (2016), 366–375. DOI 10.1016/j.topol.2016.07.021
[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. DOI 10.4064/ba8087-12-2016
[12] Keremedis K., Tachtsis E.: Compact metric spaces and weak forms of the axiom of choice. MLQ Math. Log. Q. 47 (2001), 117–128. DOI 10.1002/1521-3870(200101)47:1<117::AID-MALQ117>3.0.CO;2-N
[13] Munkres J. R.: Topology. Prentice-Hall, New Jersey, 1975. Zbl 0951.54001
[14] Tachtsis E.: Disasters in metric topology without choice. Comment. Math. Univ. Carolin. 43 (2002), no. 1, 165–174.
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