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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A cardinal inequality for topological spaces involving closed discrete sets


Authors: John Ginsburg and R. Grant Woods
Journal: Proc. Amer. Math. Soc. 64 (1977), 357-360
MSC: Primary 54A25
MathSciNet review: 0461407
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Abstract: Let X be a $ {T_1}$ topological space. Let $ a(X) = \sup \{ \alpha :X$ has a closed discrete subspace of cardinality $ \alpha \} $ and $ v(X) = \min \{ \alpha :{\Delta _X}$ can be written as the intersection of $ \alpha $ open subsets of $ X \times X\} $; here $ {\Delta _X}$ denotes the diagonal $ \{ (x,x):x \in X\} $ of X. It is proved that $ \vert X\vert \leqslant \exp (a(X)v(X))$. If, in addition, X is Hausdorff, then X has no more than $ \exp (a(X)v(X))$ compact subsets.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0461407-7
PII: S 0002-9939(1977)0461407-7
Article copyright: © Copyright 1977 American Mathematical Society