On the cardinality of the θ-closed hull of sets

The theta-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-bitightness small number of a space X, bts(0)(X), and prove that in every topological space X, the cardinality of the theta-closed hull of each set A is at most vertical bar A vertical bar(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. (C) 2013 Elsevier B.V. All rights reserved.