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|≤2wt(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|≤2at(X)πχ(X). This improves the result of Arhangel′skiĭ, van Mill, and Ridderbos in [2] that |X|≤2t(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|≤2pwcL(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)≤2L(X)wt(X)pct(X). This generalizes a result in [4] that if X is a homogeneous compactum then w(X)≤2wt(X).