On the cardinality of the $\theta$-closed hull of sets

Abstract

The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all $closed$ neighborhoods of a point intersect C, this point is in C. We define a new topological cardinal invariant function, the $\theta -bitighnesssmallnumber$ of a space X, bts_theta(X), and prove that in every topological space X, the cardinality of the theta-closed hull of each set A is at most |A|^{bts_theta(X)}. Using this result, we synthesize all earlier results on bounds on the cardinality of theta-closed hulls. We provide applications to P-spaces and to the almost-Lindelof number.

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