Cardinal Bounds in Spaces with a π-Base Whose Elements have an H-Closed Closure

Cardinal functions play a pivotal role in set-theoretic topology, offering insights into the structural properties and cardinality relationships of topological spaces. These functions serve as powerful tools for characterizing and distinguishing between different types of spaces, shedding light on fundamental questions in topology. A very interesting branch in this area is the studying of cardinal inequalities. Most of the times, these inequalities provide bounds on the cardinality of a particular class of spaces. 

We deal with the class of Hausdorff spaces having a $\pi$-base whose elements have an H-closed closure. Recently, Carlson proved that $|X|\leq 2^{wL(X)\psi_c(X)t(X)}$ for every quasiregular space $X$ with a $\pi$-base whose elements have an H-closed closure. We provide an example of a space $X$ having a $\pi$-base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that $|X|> 2^{wL(X)\chi(X)}$ (hence $|X|> 2^{wL(X)\psi_c(X)t(X)}$). In the class of spaces with a $\pi$-base whose elements have an H-closed closure, we establish the bound $|X|\leq2^{wL(X)k(X)}$ for Urysohn spaces and we give an example of an Urysohn space $Z$ such that $k(Z)<\chi(Z)$. Where $k(X)$ is a cardinal function introduced by Alas and Kocinac in 2000. Lastly, we present some characterizations of Martin's Axiom involving spaces with a $\pi$-base whose elements have an H-closed closure and, additionally, we prove that any quasiregular space having a $\pi$-base whose elements have an H-closed closure is Baire.