On closures of discrete sets

Abstract

The depth of a topological space X (g(X)) is defined as the supremum of the cardinalities of closures of discrete subsets of X. Solving a problem of Martínez-Ruiz, Ramírez-Páramo and Romero-Morales, we prove that the cardinal inequality |X| ≤ g(X) ^{L (X) F (X)} holds for every Hausdorff space X, where L(X) is the Lindelöof number of X and F (X) is the supremum of the cardinalities of the free sequences in X.