Cardinal functions $F_{\theta}(X)$ and $t_{\theta}(X)$ for H-closed spaces

TL;DR

This paper explores relationships among various cardinal functions in H-closed and Urysohn spaces, proving equalities and inequalities, and answers open questions from previous research.

Contribution

It establishes equalities among cardinal functions in H-closed and Urysohn spaces and provides counterexamples and answers to open questions.

Findings

Proved that certain cardinal functions are equal in H-closed and Urysohn spaces.

Constructed an H-closed space with a strict inequality among cardinal functions.

Answered open questions posed by Cammaroto and Kočinac.

Abstract

In this article, starting with results from a paper by Cammaroto and Ko\v{c}inac in 1993, we prove that $t_{\theta}(X) = t_{\theta}(X_{s}) = t(X_{s}) = F_{\theta}(X) = F_{\theta}(X_{s}) = F(X_{s})$ for every H-closed and Urysohn space and we present some useful examples to study the full relationships among the cardinal functions $t$, $t_{\theta}$, $F$ and $F_{\theta}$ in the context of H-closed and Urysohn spaces. Also, we construct an H-closed space $H$ for which $F_{\theta}(H) < t_{\theta}(H)$. Two questions posed by Cammaroto and Ko\v{c}inac about H-closed spaces are answered.