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

TL;DR

This paper advances the understanding of the cardinality of the $ heta$-closed hulls of sets by establishing new upper bounds based on various cardinal functions, and provides examples including a space that addresses an open question.

Contribution

It introduces new upper bounds for the cardinality of $ heta$-closed hulls using cardinal functions like $ heta$-bitightness and presents examples that include an answer to an open problem.

Findings

New upper bounds for the cardinality of $ heta$-closed hulls.

Application of cardinal functions such as $ heta$-bitightness.

An example space that answers an open question.

Abstract

The research in this paper is a continuation of the investigation of the cardinality of the $\theta$-closed hull of subsets of spaces. This research obtains new upper bounds of the cardinality of the $\theta$-closed hull of subsets using cardinal functions of $\theta$-bitightness (\cite{Ca-Ko}), finite $\theta$-bitightness (\cite{Ca-Ca-Pa-Ts}) and $\theta$-bitightness small number (\cite{Ca-Ca-Pa-Ts}) of spaces. In the final section, examples of spaces are presented including one that answers a question posed in \cite{Bo-Ca-Ma} and \cite{Bo-Pa}.