On the cardinality of the $\theta$-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 $\theta-bitighness small number$ 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-Lindelof number. 
Filippo Cammaroto, Andrei Catalioto, Bruno Antonio Pansera, Boaz   Tsaban
math.GN