ON THE CARDINALITY OF HOMOGENEOUS COMPACTA OF COUNTABLE TIGHTNESS

Abstract

We prove that every homogeneous compacta of countable tightness and d(X)≤2 ℵ o , is first countable. A relevant conjecture is raised by Arhangel'skii, conjec- ture 1.17 in (1), see also van Mill (11), which says: every homogeneous compacta of countable tightness is first countable. For all undefined notions, see Engelking(6), Kunnen(10),and Juhasz(9). Recall that πχ(X), πχ(A ), πω(X), ω(X), d(X) and t(X) denote the π − character, π − character of A,π − weight, weight,density and tightness of X. A space X is homogeneous iff for every x,y ∈ X there is a homeomorphism f of X onto X with f(x) = y. A space is hereditarily separable (HS) iff every subspace is separable. A space is power homogeneous if X k is homogeneous for some k. All spaces under discussion are Tychonoff. In this paper, we prove that if a space X is homogeneous compactum of countable tightness and d(X)≤2 ℵ o , then it is first countable. Results of the same flavour were obtained by Bell (4),and Arhangel'skii (2). Bell proved that if X is a continuous image of a compact ordered space and X is power homogeneous, then X is first countable. Arhangel'skii proved that if X is Corson compact and power homogeneous then X is first countable, and a compact scattered power homogeneous space is countable. A recently interesting result was obtained by van Mill (12). He constructed a compactum of countable π − weight and character ℵ1 with the property that it is homogeneous under MA+� CH whereas CH implies that it is not.