On the cardinality of the θ-closed hull of sets

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 θ-bitightness small number of a space X, btsθ(X), and prove that in every topological space X, the cardinality of the θ-closed hull of each set A is at most |A|btsθ(X). Using this result, we synthesize all earlier results on bounds on the cardinality of θ-closed hulls. We provide applications to P-spaces and to the almost-Lindelöf number. © 2013 Elsevier B.V.