Some Notions of Separability of Metric Spaces in ZF and Their Relation to Compactness

• Autorzy: Keremedis K.

Identyfikatory

DOI

10.4064/ba8087-12-2016

• Języki publikacji: EN

Abstrakty

EN

In the realm of metric spaces we show in ZF that: (1) Quasi separability (a metric space X = (X, d) is quasi separable iff X has a dense subset which is expressible as a countable union of finite sets) is the weakest property under which a limit point compact metric space is compact. (2) ω-quasi separability (a metric space X = (X, d) is ω-quasi separable iff X has a dense subset which is expressible as a countable union of countable sets) is a property under which a countably compact metric space is compact. (3) The statement “Every totally bounded metric space is separable” does not imply the countable choice axiom CAC.

Słowa kluczowe

EN

axiom of choice; separable; compact; countably compact; sequentially compact; complete; totally bounded metric spaces; Lebesgue metric spaces

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2016
• Tom: Vol. 64, no. 2/3
• Strony: 119--136
• Opis fizyczny: Bibliogr. 10 poz., tab.

Twórcy

autor

Keremedis K.

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

Bibliografia

• [1] C. Good and I. J. Tree, Continuing horrors of topology without choice, Topology Appl. 63 (1995), 79–90.
• [2] G. Gutierres, Total boundedness and the axiom of choice, Appl. Categor. Structures 24 (2016), 457–469.
• [3] P. Howard, K. Keremedis, J. E. Rubin, A. Stanley and E. Tachtsis, Non-constructive properties of the real line, Math. Logic Quart. 47 (2001), 423–431.
• [4] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998.
• [5] T. Jech, The Axiom of Choice, North-Holland, 1973.
• [6] K. Keremedis, On the relative strength of forms of compactness of metric spaces and their countable productivity in ZF, Topology Appl. 159 (2012), 3396–3403.
• [7] K. Keremedis, On sequentially compact and related notions of compactness of metric spaces in ZF, Bull. Polish Acad. Sci. Math. 64 (2016), 29–46.
• [8] K. Keremedis, On Weierstrass compact pseudometric spaces and a weak form of the axiom of choice, Topology Appl. 108 (2000), 75–78.
• [9] J. R. Munkres, Topology, Prentice-Hall, Englewood Cliffs, NJ, 1975.
• [10] J. Nagata, Modern General Topology, North-Holland, 1985.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.