On spaces with a π-base with elements with compact closure

In this paper we show that if X is a T1-space with a pi-base whose elements have compact closure, then d(X) <= c(X) <middle dot> 2(psi(X)) and therefore, for such spaces we have d(X)(psi(X)) = c(X)(psi(X)). This result allows us to restate several known upper bounds of the cardinality of a Hausdorff space X by replacing in them d(X) with c(X). In addition, we show that for such spaces X Sapirovskil's inequality d(X) <= pi chi(X)(c(X)), which is known to be true for regular Hausdorff spaces, is also valid. In the case when the space X is in addition sequential or radial, we show that |X| <= 2(c(X)). This result extends two theorems of Arhangeliskil to the class of Hausdorff spaces with a pi-base whose elements have compact closures. We also show that spaces with a pi-base with elements with compact closures are alpha-favorable in the Banach- Mazur game, which implies such spaces are Baire. It was shown in Bella et al. (Quaest Math 46(4):745-760, 2023) that if a Hausdorff space X has a pi-base consisting of elements with compact closure, then |X| <= 2(wL(X)t(X)psi c(X)). We give a variation of this result by showing |X| < pi chi((X)wL(X)ot(X)psi c(X)) for such a space X. Note that since wL(X)ot(X) <= c(X), this result is at least as good as that given by Sun (Proc Am Math Soc 104:313-316, 1988). We also give a possible improvement of the bound in Bella et al. (2023) by showing that |X| <= 2(wL(X)wt(X)psi c(X)) for a Hausdorff space X with a pi-base consisting of elements with compact closure. This uses the weak tightness wt(X) defined in Carlson (Topol Appl 249:103-111, 2018), which has the property ot(X) <= wt(X) <= t(X). We also show that if X is a Hausdorff homogeneous space with a pi-base consisting of elements with compact closure (such spaces are locally compact), then |X| < wL(X)(wt(X)pi chi(X)). This generalizes the result in Bella and Carlson (Monatsh Math 192(1):39-48, 2020) that if X is a homogeneous compactum, then |X| <= 2(wt(X)pi chi (X)).