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 $|X|\le g(X{)}^{L(X)\cdot F(X)}$ holds for every Hausdorff space $X$, where $L(X)$ is the Lindelöf number of $X$ and $F(X)$ is the supremum of the cardinalities of the free sequences in $X$.