On the cardinality of power homogeneous Hausdorff spaces

Abstract

Abstract. We prove that the cardinality of power homogeneous Hausdorff spaces X is bounded by d(X) πχ(X) . This inequality improves many known results and it also solves a question by J. van Mill. We further introduce ∆-power homogeneity, which leads to a new proof of van Douwen's theorem.1. Introduction. A space X is homogeneous if for every x, y ∈ X there is a homeomorphism h of X such that h(x) = y. A space X is called power homogeneous if X µ is homogeneous for some cardinal number µ. By πχ(X) and πw(X) we denote the π-character and π-weight respectively. By d(X), w(X), ψw(X) and c(X) we denote density, weight, pseudo-weight and cellularity.In 1978, E. van Douwen proved in [4] that the cardinality of power homogeneous Hausdorff spaces X is bounded by 2 πw(X) . Applying results of Shapirovskiȋ [12] and Ismail [6], A. V. Arkhangel ′ skiȋ noted in [1, Theorem 1.5] that the cardinality of homogeneous regular spaces X is bounded by 2 πχ(X)c(X) . Since always πχ(X)c(X) ≤ πw(X) and strict inequality is possible, this result improves van Douwen's theorem for the class of homogeneous spaces. Recently, J. van Mill [8] extended Arkhangel ′ skiȋ's result to the class of power homogeneous compacta. In his paper van Mill asks whether this result can also be proved for power homogeneous spaces without the assumption of compactness (see [8, Remark 2.7]). A partial answer to this question was provided by A. Bella in [3]. In the present paper we provide a full positive answer to van Mill's question (see Corollary 3.5).In [4] van Douwen studies power homogeneous spaces by looking at the number of possible ways certain sequences of open sets cluster at points. This same method was applied by van Mill in [8]. In the present paper we introduce an entirely different technique which follows from results in [2].

On cardinality bounds involving the weak Lindelöf degree

“…To this end we show that if U ⊆ clD ⊆ X, where X is power homogeneous and U is open, then |U | ≤ |D| πχ(X) . This is a strengthening of a result of Ridderbos [18].…”

“…For a subset C ⊆ µ, let π C : X µ → X C be the projection, and for x ∈ X let x C be the point π C (x) ∈ X C . We also adopt the notation given before and after Theorem 3.1 in [18], which features heavily in the next proof. Lemma 3.3.…”

“…(De la Vega first established the bound 2 t(X) for compact homogeneous spaces in [9]). A key ingredient is Lemma 3.4, which slightly improves Ridderbos' bound d(X) πχ(X) [18] for the cardinality of a power homogeneous space X and uses modified techniques from that paper. The cardinality bound 2 wL(X)t(X) for locally compact, power homogeneous spaces is then a "companion bound" to the bound 2 wL(X)ψ(X) for general locally compact spaces, both fundamentally proved with the same closing-off argument given in Theorem 2.1.…”

On the weak tightness and power homogeneous compacta

“…It was in fact generalized in [5]: if X is power homogeneous, D ⊆ X, and U is an open set such that U ⊆ D, then |U | ≤ |D| πχ(X) . Lemma 4.5 (Ridderbos [15]). If X is power homogeneous and Hausdorff then |X| ≤ d(X) χ(X) .…”

Power homogeneous compacta and the order theory of local bases

“…• |X| d( X) πχ(X) for every power homogeneous Hausdorff X [19]. Theorem 1.3's cardinality bound of 2 πχ(X)c(X) was used in the proof of Theorem 1.7, so it is natural to ask to what extent Theorem 1.7 is true of power homogeneous compacta, which satisfy the same cardinality bound.…”