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Abstrakty
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.
Wydawca
Rocznik
Tom
Strony
99--113
Opis fizyczny
Bibliogr. 23 poz., rys.
Twórcy
autor
- Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece
Bibliografia
- [1] R. W. Bagley, E. H. Connell, and J. D. McKnight, Jr., On properties characterizing pseudocompact spaces, Proc. Amer. Math. Soc. 9 (1958), 500-506.
- [2] N. Brunner, Lindelöf Räume und Auswahlaxiom, Anz. Österr. Akad. Wiss. Math.-Nat. Kl. 119 (1982), 161-165.
- [3] R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.
- [4] C. Good, I. J. Tree and W. S. Watson, On Stone’s theorem and the axiom of choice, Proc. Amer. Math. Soc. 126 (1998), 1211-1218.
- [5] P. Howard, K. Keremedis, H. Rubin and J. E. Rubin, Versions of normality and some weak forms of the axiom of choice, Math. Logic Quart. 44 (1998), 367-382.
- [6] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998.
- [7] K. Keremedis, On metric spaces where continuous real valued functions are uniformy continuous in ZF, Topology Appl. 210 (2016), 366-375.
- [8] K. Keremedis, Some notions of separability of metric spaces in ZF and their relation to compactness, Bull. Polish Acad. Sci. Math. 64 (2016), 109-136.
- [9] 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.
- [10] K. Keremedis, Non-discrete metrics in ZF and some notions of finiteness, Math. Logic Quart. 62 (2016), 383-390.
- [11] K. Keremedis, On lightly and countably compact spaces in ZF, Quaestiones Math. (to appear).
- [12] K. Keremedis and E. Tachtsis, On Loeb and weakly Loeb Hausdorff spaces, Sci. Math. Japon. 53 (2001), 247-251.
- [13] A. Lévy, The independence of various definitions of finiteness, Fund. Math. 46 (1958), 1-13.
- [14] S. Mardešic et P. Papić, Sur les espaces dont toute transformation réelle continue Est bornée, Hrvatsko Prirod. Društvo Glasnik Mat.-Fiz. Astr. Ser. II 10 (1955), 225-232.
- [15] S. Mrówka, On normal metrics, Amer. Math. Monthly 72 (1965), 998-1001.
- [16] J. R. Munkres, Topology, Prentice-Hall, Englewood Cliffs, NJ, 1975.
- [17] C. Ryll-Nardzewski, A remark on the Cartesian product of two compact spaces, Bull. Acad. Polon. Sci. Cl. III 2 (1954), 265-266.
- [18] M. E. Rudin, A new proof that metric spaces are paracompact, Proc. Amer. Math. Soc. 20 (1969), 603.
- [19] A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948), 977-982.
- [20] A. H. Stone, Hereditarily compact spaces, Amer. J. Math. 82 (1960), 900-914.
- [21] E. Tachtsis, Disasters in metric topology without choice, Comment. Math. Univ. Carolin. 43 (2002), 165-174.
- [22] A. Tarski, Sur les ensembles finis, Fund. Math. 6 (1924), 45-95.
- [23] S. W. Watson, Pseudocompact metacompact spaces are compact, Proc. Amer. Math. Soc. 81 (1981), 151-152.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
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Bibliografia
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