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

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 $\mathrm{\mid }X\mathrm{\mid }\le 2^{L(X)wt(X)\psi (X)}$ for any Hausdorff space $X$ (Theorem 2.8). As $wt(X)\le t(X)$ for any space $X$, this generalizes the well-known cardinal inequality $\mathrm{\mid }X\mathrm{\mid }\le 2^{L(X)t(X)\psi (X)}$ 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 $\mathrm{\mid }X\mathrm{\mid }\le \mathfrak{c}$, the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.