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

Abstract

We deal with the class of Hausdorff spaces having a $\pi$-base whose elements have an H-closed closure. 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 $|X|> 2^{wL(X)\chi(X)}$ (then $|X|> 2^{wL(X)\psi_c(X)t(X)}$). Still 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 $k(Z)<\chi(Z)$. Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a $\pi$-base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a $\pi$-base whose elements have an H-closed closure then such space is Baire.