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 | References | Similar Articles | Additional Information

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