A cardinal inequality for topological spaces involving closed discrete sets

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.