On closures of discrete sets

Abstract

The depth of a topological space X (g(X)) is dened as the supremum of the cardinalities of closures of discrete subsets of X. Solving a problem of Martnez-Ruiz, Ramrez-Paramo and Romero-Morales, we prove that the cardinal inequality lX∣ ≤ g(X)^{L(X)F (X)} holds for every Hausdorff space X, where L(X) is the Lindelof number of X and F (X) is the supremum of the cardinalities of the free sequences in X.