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].

We show that the cardinality of power homogeneous T5 compacta X is bounded by 2 c ( X ) . This answers a question of J. van Mill, who proved this bound for homogeneous T5 compacta. We further extend some results of I. Juhász, P. Nyikos and Z. Szentmiklóssy and as a corollary we prove that consistently every power homogeneous T5 compactum is first countable. This improves a theorem of R. de la Vega who proved this consistency result for homogeneous T5 compacta.

Improving on an earlier example by J. van Mill, we prove that there exists a zero-dimensional compact space of countable pi-weight and uncountable character which is homogeneous under MA+notCH, but not under CH.Comment: Revision after referee's comment

No abstract

No abstract