Power homogeneous compacta and variations on tightness

Abstract

The weak tightness wt(X), introduced in [6], has the property wt(X) ≤ t(X). It was shown in [4] that if X is a homogeneous compactum then |X| ≤ 2 wt(X)πχ(X) . We introduce the almost tightness at(X) with the property wt(X) ≤ at(X) ≤ t(X) and show that if X is a power homogeneous compactum then |X| ≤ 2 at(X)πχ(X) . This improves the result of Arhangel ′ skiȋ, van Mill, and Ridderbos in [2] that |X| ≤ 2 t(X) for a power homogeneous compactum X and gives a partial answer to a question in [4]. In addition, if X is a homogeneous Hausdorff space we show that |X| ≤ 2 pwc L(X)wt(X)πχ(X)pct(X) , improving a result in [3]. It also extends the result in [4] into the Hausdorff setting. The cardinal invariant pwLc(X), introduced in [5] by Bella and Spadaro, satisfies pwLc(X) ≤ L(X) and pwLc(X) ≤ c(X). We also show the weight w(X) of a homogeneous space X is bounded in various contexts using wt(X). One such result is that if X is homogeneous and regular then w(X) ≤ 2 L(X)wt(X)pct(X) . This generalizes a result in [4] that if X is a homogeneous compactum then w(X) ≤ 2 wt(X) .