On spaces with a \(\pi\)-base whose elements have an H-closed closure

We deal with the class of Hausdorff spaces having a \(\pi\) $\pi$ -base whose elements have an H-closed closure. Carlson proved that \(|X|\leq 2^{wL(X)\psi_c(X)t(X)}\) $|X|\le 2^{wL(X){\psi }_c(X)t(X)}$ for every quasiregular space \(X\) $X$ with a \(\pi\) $\pi$ -base whose elements have an H-closed closure. We provide an example of a space \(X\) $X$ having a \(\pi\) $\pi$ -base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that \(|X|> 2^{wL(X)\chi(X)}\) $|X|>2^{wL(X)\chi (X)}$ (then \(|X|> 2^{wL(X)\psi_c(X)t(X)}\) $|X|>2^{wL(X){\psi }_c(X)t(X)}$ ). Always in the class of spaces with a \(\pi\) $\pi$ -base whose elements have an H-closed closure, we establish the bound \(|X|\leq2^{wL(X)k(X)}\) $|X|\le 2^{wL(X)k(X)}$ for Urysohn spaces and we give an example of an Urysohn space \(Z\) $Z$ such that \(k(Z)<\chi(Z)\) $k(Z)<\chi (Z)$ . Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a \(\pi\) $\pi$ -base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a \(\pi\) $\pi$ -base whose elements have an H-closed closure then such a space is Baire.