Power homogeneous compacta and variations on tightness

TL;DR

This paper introduces the almost tightness invariant and explores its implications for bounding the cardinality of power homogeneous and homogeneous spaces, extending and improving existing results in topology.

Contribution

It defines the almost tightness $at(X)$ and demonstrates its use in establishing sharper bounds on the size of certain homogeneous spaces, extending prior results.

Findings

Bound $|X| ext{ for power homogeneous compacta using } at(X)

Improved bounds on $|X|$ for homogeneous Hausdorff spaces

Bound on weight $w(X)$ for homogeneous regular spaces

Abstract

The weak tightness $wt(X)$, introduced in [6], has the property $wt(X)\leq t(X)$. It was shown in [4] that if $X$ is a homogeneous compactum then $|X|\leq 2^{wt(X)\pi\chi(X)}$. We introduce the almost tightness $at(X)$ with the property $wt(X)\leq at(X)\leq t(X)$ and show that if $X$ is a power homogeneous compactum then $|X|\leq 2^{at(X)\pi\chi(X)}$. This improves the result of \arhangelskii, van Mill, and Ridderbos in [2] that $|X|\leq 2^{t(X)}$ 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 $|X|\leq 2^{pw_cL(X)wt(X)\pi\chi(X)pct(X)}$, improving a result in [3]. It also extends the result in [4] into the Hausdorff setting. The cardinal invariant $pwL_c(X)$, introduced in [5] by Bella and Spadaro, satisfies $pwL_c(X)\leq L(X)$ and $pwL_c(X)\leq c(X)$. 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 $w(X)\leq 2^{L(X)wt(X)pct(X)}$. This generalizes a result in [4] that if $X$ is a homogeneous compactum then $w(X)\leq 2^{wt(X)}$.