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

TL;DR

This paper introduces a new topological invariant, the theta-bitighness small number, and proves it bounds the cardinality of theta-closed hulls in any space, unifying previous results and applying to P-spaces and almost-Lindelof spaces.

Contribution

The paper defines the theta-bitighness small number and establishes it as a key bound for the cardinality of theta-closed hulls, synthesizing prior bounds and extending to specific classes.

Findings

Cardinality of theta-closed hulls is at most |A|^{bts_theta(X)}.

Introduces the theta-bitighness small number as a new invariant.

Provides applications to P-spaces and almost-Lindelof spaces.

Abstract

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.