On the weak tightness, Hausdorff spaces, and power homogeneous compacta

TL;DR

This paper introduces the weak tightness invariant for topological spaces, generalizes known cardinal inequalities for Hausdorff spaces, and extends results on the size of power homogeneous compacta with specific covers.

Contribution

It defines the weak tightness $wt(X)$, generalizes cardinal bounds for Hausdorff spaces, and extends size bounds for power homogeneous compacta with dense, countably tight covers.

Findings

Established $|X| extless 2^{L(X)wt(X)\psi(X)}$ for Hausdorff spaces.

Generalized bounds using $G_\kappa$-set covers.

Proved that certain power homogeneous compacta have size at most continuum.

Abstract

Motivated by results of Juh\'asz and van Mill in [13], we define the cardinal invariant $wt(X)$, the weak tightness of a topological space $X$, and show that $|X|\leq 2^{L(X)wt(X)\psi(X)}$ for any Hausdorff space $X$ (Theorem 2.8). As $wt(X)\leq t(X)$ for any space $X$, this generalizes the well-known cardinal inequality $|X|\leq 2^{L(X)t(X)\psi(X)}$ for Hausdorff spaces (Arhangel{\cprime}ski\u{i}~[1],\v{S}}apirovski\u{i}}~[18]) in a new direction. Theorem 2.8 is generalized further using covers by $G_\kappa$-sets, where $\kappa$ is a cardinal, to show that if $X$ is a power homogeneous compactum with a countable cover of dense, countably tight subspaces then $|X|\leq\mathfrak{c}$, the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.