On the cardinality of the theta-closed hull of set

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-bitighness small number 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-Lindelöf number.