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Proof of a theorem of Burke and Hodel on the cardinality of topological spaces

Author: Robert L. Blair
Journal: Proc. Amer. Math. Soc. 78 (1980), 449-450
MSC: Primary 54A25
MathSciNet review: 553393
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Abstract: Techniques of Pol are used to give a direct proof of Burke and Hodel's inequality $ \vert X\vert \leqslant {2^{\Delta (X) \cdot {\text{psw}}(X)}}$, where $ \Delta (X)$ is the discreteness character of the $ {T_1}$ space X and $ {\text{psw}}(X)$ is the point separating weight of X.

References [Enhancements On Off] (What's this?)

  • [BH] D. K. Burke and R. E. Hodel, The number of compact subsets of a topological space, Proc. Amer. Math. Soc. 58 (1976), 363-368. MR 0418014 (54:6058)
  • [E] R. Engelking, General topology, PWN, Warsaw, 1975; English Transl., PWN, Warsaw, 1977. MR 0500780 (58:18316b)
  • [P$ _{1}$] R. Pol, Short proof of two theorems on cardinality of topological spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 1245-1249. MR 0383333 (52:4214)
  • [P$ _{2}$] V. I. Ponomarev, On the cardinality of bicompacta which satisfy the first axiom of countability, Dokl. Akad. Nauk SSSR 196 (1971), 296-298 = Soviet Math. Dokl. 12 (1971), 121-124. MR 0275365 (43:1122)
  • [S] B. Šapirovskiĭ, On discrete subspaces of topological spaces: weight, tightness and Suslin number, Dokl. Akad. Nauk SSSR 202 (1972), 779-782 = Soviet Math. Dokl. 13 (1972), 215-219. MR 0292012 (45:1100)

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Keywords: Cardinality of a topological space, separating open cover, point separating weight, discreteness character
Article copyright: © Copyright 1980 American Mathematical Society

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