A cardinal inequality for topological spaces involving closed discrete sets

Abstract

Let X be a T 1 {T_1} topological space. Let a ( X ) = sup { α : X a(X) = \sup \{ \alpha :X has a closed discrete subspace of cardinality α } \alpha \} and v ( X ) = min { α : Δ X v(X) = \min \{ \alpha :{\Delta _X} can be written as the intersection of α \alpha open subsets of X × X } X \times X\} ; here Δ X {\Delta _X} denotes the diagonal { ( x , x ) : x ∈ X } \{ (x,x):x \in X\} of X. It is proved that | X | ⩽ exp ⁡ ( a ( X ) v ( X ) ) |X| \leqslant \exp (a(X)v(X)) . If, in addition, X is Hausdorff, then X has no more than exp ⁡ ( a ( X ) v ( X ) ) \exp (a(X)v(X)) compact subsets.