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

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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 (Arhangelskiĭ [1],Šapirovskiĭ [18]) 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|, the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.