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 Lindelöf number of X and F(X) is the supremum of the cardinalities of the free sequences in X.
S. Spadaro (2021). On closures of discrete sets. QUAESTIONES MATHEMATICAE, 44(6), 717-720 [10.2989/16073606.2019.1617364].
On closures of discrete sets
S. Spadaro
2021-01-01
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 Lindelöf number of X and F(X) is the supremum of the cardinalities of the free sequences in X.
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