On several cardinality bounds on power homogeneous spaces

Abstract. We show the cardinality of a homogeneous Hausdorff space X is not necessarily bounded by 2L(X)piχ(X) by providing examples of σ-compact, countably tight, homogeneous spaces of countable pi-character and arbitrary cardinality. We also generalize a closing-off argument of Pytkeev to show the cardinality of any power homogeneous Hausdorff space X is at most 2L(X)pct(X)t(X). This was previously shown to hold if X is also regular by G.J. Ridderbos. Another consequence of the generalization of Pytkeev’s closing-off argument is the well-known cardinality bound 2L(X)t(X)ψ(X) for an arbitrary Hausdorff space X. 1. Overview In this study we investigate several cardinality bounds on power homogeneous topological spaces. Recall that a space X is homogeneous if for every x, y ∈ X there is a homeomorphism h of X such that h(x) = y. That is, in essence, all points in X “look the same ” topologically in the sense that all neighborhood filters are translates of each other under homeomorphisms. A space X is power homogeneous if Xµ is homogeneous for some cardinal µ. We use several cardinal invariants throughout this work. By L(X), w(X), nw(X), χ(X), piχ(X), piw(X), d(X), c(X), and t(X) we denote the Lindelöf degree, weight, network weight, character, pi-character, pi-weight, density, cellularity, and tightness, respectively, of a space X. See [7] for all undefined notions. All spaces under consideration ar