Countably compact, locally countable $T_{2}$-spaces
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- by J. E. Vaughan PDF
- Proc. Amer. Math. Soc. 80 (1980), 147-153 Request permission
Abstract:
The results of this paper provide a simple method for constructing locally countable ${T_2}$-spaces (not ${T_3}$) in which every infinite closed set has cardinality ${2^c}$. The spaces are used in a variety of ways as counterexamples. One of these spaces may be considered as a countably compact version of the Katětov H-closed extension of the natural numbers.References
- C. E. Aull, A certain class of topological spaces, Prace Mat. 11 (1967), 49–53. MR 0227914
- Allen R. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1969/70), 185–193. MR 251697, DOI 10.4064/fm-66-2-185-193
- W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267, DOI 10.1007/978-3-642-65780-1
- Eric K. van Douwen, The Čech-Stone remainder of some nowhere locally compact spaces, Topology Appl. 47 (1992), no. 3, 165–171. MR 1192306, DOI 10.1016/0166-8641(92)90027-W
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- John Ginsburg and Victor Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), no. 2, 403–418. MR 380736, DOI 10.2140/pjm.1975.57.403 A. Hajnal and I. Juhász, Some remarks on a property of topological cardinal functions, Acta Math. Acad. Sci. Hungar. 24 (1973), 307-312.
- R. E. Hodel, The number of closed subsets of a topological space, Canadian J. Math. 30 (1978), no. 2, 301–314. MR 464131, DOI 10.4153/CJM-1978-027-7
- I. Juhász, Cardinal functions in topology, Mathematical Centre Tracts, No. 34, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg. MR 0340021 I. Juhász, B. Sz.-Nagy and W. Weiss, On countably compact, locally countable spaces (preprint).
- D. J. Lutzer, Semimetrizable and stratifiable spaces, General Topology and Appl. 1 (1971), no. 1, 43–48. MR 296893, DOI 10.1016/0016-660X(71)90109-7
- J. Novák, On the Cartesian product of two compact spaces, Fund. Math. 40 (1953), 106–112. MR 60212, DOI 10.4064/fm-40-1-106-112
- M. Rajagopalan and R. Grant Woods, Products of sequentially compact spaces and the $V$-process, Trans. Amer. Math. Soc. 232 (1977), 245–253. MR 451219, DOI 10.1090/S0002-9947-1977-0451219-7
- C. Ryll-Nardzewski, A remark on the Cartesian product of two compact spaces, Bull. Acad. Polon. Sci. Cl. III. 2 (1954), 265–266. MR 0063652
- J. E. Vaughan, Products of perfectly normal, sequentially compact spaces, J. London Math. Soc. (2) 14 (1976), no. 3, 517–520. MR 464163, DOI 10.1112/jlms/s2-14.3.517
- J. E. Vaughan, Discrete sequences of points, Topology Proc. 3 (1978), no. 1, 237–265 (1979). MR 540494
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 147-153
- MSC: Primary 54D20; Secondary 54A25, 54D25, 54D30, 54G20
- DOI: https://doi.org/10.1090/S0002-9939-1980-0574525-X
- MathSciNet review: 574525