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

We deal with the class of Hausdorff spaces having a π-base whose elements have an H-closed closure. Carlson proved that |X| ≤ 2 wL(X)ψc(X)t(X) for every quasiregular space X with a π-base whose elements have an H-closed closure. We provide an example of a space X having a π-base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that |X| > 2 wL(X)χ(X) (then |X| > 2 wL(X)ψc(X)t(X) ). Always in the class of spaces with a π-base whose elements have an H-closed closure, we establish the bound |X| ≤ 2 wL(X)k(X) for Urysohn spaces and we give an example of an Urysohn space Z such that k(Z) < χ(Z). Lastly, we present some equivalent conditions to the Martin’s Axiom involving spaces with a π-base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a π-base whose elements have an H-closed closure then such a space is Baire.