On pseudocompactness and light compactness of metric spaces in ZF

• Autorzy: Keremedis K.

Identyfikatory

DOI

10.4064/ba8131-10-2018

• Języki publikacji: EN

Abstrakty

EN

In the realm of metric spaces we show, in the Zermelo-Fraenkel set theory ZF, that: (a) A metric space X = (X, d) is countably compact iff it is pseudocompact. (b) Given a metric space X = (X, d); the following statements are equivalent: (i) X is lightly compact (every locally finite family of open sets is finite). (ii) Every locally finite family of subsets of X is finite. (iii) Every locally finite family of closed subsets of X is finite. (iv) Every pairwise disjoint, locally finite family of subsets of X is finite. (v) Every pairwise disjoint, locally finite family of closed subsets of X is finite. (vi) Every locally finite, pairwise disjoint family of open subsets of X is finite. (vii) Every locally finite open cover of X has a finite subcover. (c) For every infinite set X, the powerset P(X) of X has a countably infinite subset iff every countably compact metric space is lightly compact.

Słowa kluczowe

EN

axiom of choice; countably compact metric spaces; pseudocompact metric spaces

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 2
• Strony: 99--113
• Opis fizyczny: Bibliogr. 23 poz., rys.

Twórcy

autor

Keremedis K.

• Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece

Bibliografia

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Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).