On spaces with a π-base whose elements have an H-closed closure

We deal with the class of Hausdorff spaces having a pi-base whose elements have an H-closed closure. Carlson proved that |X| <= 2(wL(X)psi c(X)t(X) )for every quasiregular spaceX with a pi-base whose elements have an H-closed closure. We provide an example of a spaceX 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|>(2wL(X)psi c(X)t(X))). Always in the class of spaces with a pi-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 Zsuch 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 a space is Baire.