Point-countability and compactness
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- by H. H. Wicke and J. M. Worrell PDF
- Proc. Amer. Math. Soc. 55 (1976), 427-431 Request permission
Abstract:
We prove that if a countably compact space $X$ has an open cover $\mathcal {U} = \bigcup {\{ {\mathcal {V}_n}:n < \omega \} }$ such that each $x \in X$ is in at least one but not more than countably many elements of some ${\mathcal {V}_n}$, then some finite subcollection of $\mathcal {U}$ covers $X$. We apply the theorem in proving several metrization theorems for countably compact spaces and discuss consequences of weak $\delta \theta$-refinability, a concept implicit in the statement of the theorem.References
-
G. Aquaro, Point countable coverings in countably compact spaces, General Topology and its Relations to Modern Analysis and Algebra, II (Prague, 1967), pp. 39-41.
- Richard Arens and J. Dugundji, Remark on the concept of compactness, Portugal. Math. 9 (1950), 141–143. MR 38642
- C. E. Aull, A generalization of a theorem of Aquaro, Bull. Austral. Math. Soc. 9 (1973), 105–108. MR 372817, DOI 10.1017/S0004972700042933
- Harold R. Bennett, On quasi-developable spaces, General Topology and Appl. 1 (1971), no. 3, 253–262. MR 288725
- H. R. Bennett and D. J. Lutzer, A note on weak $\theta$-refinability, General Topology and Appl. 2 (1972), 49–54. MR 301697
- Dennis K. Burke, On $p$-spaces and $w\Delta$-spaces, Pacific J. Math. 35 (1970), 285–296. MR 278255
- Geoffrey D. Creede, Concerning semi-stratifiable spaces, Pacific J. Math. 32 (1970), 47–54. MR 254799
- Jean Dieudonné, Une généralisation des espaces compacts, J. Math. Pures Appl. (9) 23 (1944), 65–76 (French). MR 13297
- J. Donald Monk, Introduction to set theory, McGraw-Hill Book Co., New York-London-Sydney, 1969. MR 0286668
- R. L. Moore, Foundations of point set theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722 M. E. Rudin, Lecture notes, Univ. of Wyoming Topology Conf. (Laramie, Wy., 1974).
- Howard H. Wicke, Open continuous images of certain kinds of $M$-spaces and completeness of mappings and spaces, General Topology and Appl. 1 (1971), no. 1, 85–100. MR 282348 —, On spaces whose diagonal is a set of interior condensation, Notices Amer. Math. Soc. 19(1972), A-657. Abstract #696-54-16. H. H. Wicke and J. M. Worrell, Jr., Characterizations of absolute sets of interior condensation, Notices Amer. Math. Soc. 20(1973), A-401. Abstract #704-G6.
- H. H. Wicke and J. M. Worrell Jr., Topological completeness of first countable Hausdorff spaces. II, Fund. Math. 91 (1976), no. 1, 11–27. MR 461457, DOI 10.4064/fm-91-1-11-27
- H. H. Wicke and J. M. Worrell Jr., A characterization of primitive bases, Proc. Amer. Math. Soc. 50 (1975), 443–450. MR 425922, DOI 10.1090/S0002-9939-1975-0425922-2
- H. H. Wicke and J. M. Worrell Jr., Primitive structures in general topology, Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), Academic Press, New York, 1975, pp. 581–599. MR 0375242 J. M. Worrell, Jr., A perfect mapping not preserving the $p$-space property, presented at Pittsburgh Conf. on General Topology, 1970.
- J. M. Worrell Jr. and H. H. Wicke, Characterizations of developable topological spaces, Canadian J. Math. 17 (1965), 820–830. MR 182945, DOI 10.4153/CJM-1965-080-3 —, Remarks on a property of $\theta$-refinable spaces, Notices Amer. Math. Soc. 21(1974), A-622. Abstract #717-G16.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 427-431
- DOI: https://doi.org/10.1090/S0002-9939-1976-0400166-X
- MathSciNet review: 0400166