On the weak tightness and power homogeneous compacta

Abstract

Motivated by results of Juhász and van Mill in [13], we define the cardinal invariant wt(X), the weak tightness of a topological space X, and show that |X|≤2L(X)wt(X)ψ(X) for any Hausdorff space X (Theorem 2.8). As wt(X)≤t(X) for any space X, this generalizes the well-known cardinal inequality |X|≤2L(X)t(X)ψ(X) for Hausdorff spaces (Arhangel′skiĭ [1], Šapirovskiĭ [17]) in a new direction. Theorem 2.8 is generalized further using covers by Gκ-sets, where κ is a cardinal, to show that if X is a power homogeneous compactum with a countable cover of dense, countably tight subspaces then |X|≤c, the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.