2021/04/02 by Carlson, Nathan
#FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2104.01273
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 \arhangelskii, 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).