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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
DOI: https://doi.org/10.1090/S0002-9939-1977-0461407-7
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 $|X| \leqslant \exp (a(X)v(X))$. If, in addition, X is Hausdorff, then X has no more than $\exp (a(X)v(X))$ compact subsets.


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Article copyright: © Copyright 1977 American Mathematical Society