On Sequential Compactness and Related Notions of Compactness of Metric Spaces in ZF

• Autorzy: Keremedis K.

Identyfikatory

DOI

10.4064/ba8054-5-2016

• Języki publikacji: EN

Abstrakty

EN

We show that: (i) If every sequentially compact metric space is countably compact then for every infinite set X, [X]^<ω is Dedekind-infinite. In particular, every infinite subset of R is Dedekind-infinite. (ii) Every sequentially compact metric space is compact iff every sequentially compact metric space is separable. In addition, if every sequentially compact metric space is compact then: every infinite set is Dedekind-infinite, the product of a countable family of compact metric spaces is compact, and every compact metric space is separable. (iii) The axiom of countable choice implies that every sequentially bounded metric space is totally bounded and separable, every sequentially compact metric space is compact, and every uncountable sequentially compact, metric space has size |R|. (iv) If every sequentially bounded metric space is totally bounded then every infinite set is Dedekind-infinite. (v) The statement: “Every sequentially bounded metric space is bounded” implies the axiom of countable choice restricted to the real line. (vi) The statement: “For every compact metric space X either |X| ≤ |R|, or |R| ≤ |X|” implies the axiom of countable choice restricted to families of finite sets. (vii) It is consistent with ZF that there exists a sequentially bounded metric space whose completion is not sequentially bounded. (viii) The notion of sequential boundedness of metric spaces is countably productive.

Słowa kluczowe

EN

weak forms of the axiom of choice; countable Tychonoff products; compact; countably compact; sequentially compact; complete; bounded; sequentially bounded; totally bounded metric spaces

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2016
• Tom: Vol. 64, no. 1
• Strony: 29--46
• Opis fizyczny: Bibliogr. 13 poz., rys.

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] 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.
• [3] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998.
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• [5] 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.
• [6] K. Keremedis, On sequentially closed subsets of the real line in ZF, Math. Logic Quart. 61 (2015), 24-35.
• [7] K. Keremedis and E. Tachtsis, Compact metric spaces and weak forms of the axiom of choice, Math. Logic. Quart. 47 (2001), 117-128.
• [8] K. Keremedis and E. Tachtsis, On sequentially compact subspaces of R without the axiom of choice, Notre Dame J. Formal Logic 44 (2003), 175-184.
• [9] J. R. Munkres, Topology, Prentice-Hall, 1975.
• [10] J. Nagata, Modern General Topology, North-Holland, 1985.
• [11] E. Tachtsis, Disasters in metric topology without choice, Comment. Math. Univ. Carolin. 43 (2002), 165-174.
• [12] E. K. van Douwen, Horrors of topology without AC: A nonnormal orderable space, Proc. Amer. Math. Soc. 95 (1985), 101-105.
• [13] S. Willard, General Topology, Addison-Wesley, 1970.

Uwagi

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