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A theorem on the cardinality of $ \kappa $-total spaces


Author: R. M. Stephenson
Journal: Proc. Amer. Math. Soc. 89 (1983), 367-370
MSC: Primary 54A25; Secondary 54D10, 54G20
DOI: https://doi.org/10.1090/S0002-9939-1983-0712653-9
MathSciNet review: 712653
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Abstract: Throughout this article, $ \kappa $ denotes an arbitrary infinite cardinal number. In 1979, A. A. Gryzlov strengthened a well-known result of A. V. Arhangel'skii by proving that every compact $ {T_1}$-space of pseudocharacter $ \kappa $ has cardinality $ \leqslant {2^\kappa }$. Using techniques similar to Gryzlov's, we prove that every $ {2^\kappa }$-total, $ {T_1}$-space of pseudocharacter $ \leqslant \kappa $ is compact and hence of cardinality $ \leqslant {2^\kappa }$. Some related results and examples are given.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0712653-9
Keywords: Initially $ \kappa $-compact spaces, $ \kappa $-total spaces
Article copyright: © Copyright 1983 American Mathematical Society

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