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

2017/09/23 by Carlson, Nathan
#54A25 #54B10 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1709.07998

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|≤ 2^{L(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|≤ 2^{L(X)t(X)ψ(X)} for Hausdorff spaces (Arhangel\cprimeskiĭ~[1],Šapirovskiĭ~[18]) in a new direction. Theorem 2.8 is generalized further using covers by G_κ-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|≤\mathfrakc, the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.

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