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.

Mathematics Subject Classification (2010):

• Primary: 54A25
• Secondary: 54D20

Key words:

• Cardinal inequality
• Lindelöf
• discrete set
• depth
• elementary submodel