On power homogeneous spaces

Abstract

The notion of a Moscow space [A.V. Arhangel'skii, Comment. Math. Univ. Carolinae 24 (1983) 105–120] and a new notion of a weakly Klebanov space, introduced below, are applied to study topological properties of spaces X such that some power of X is homogeneous. Under this approach, a crucial role is played by a theorem of Y. Yajima [J. Math. Soc. Japan 36 (1984) 689–699]. Several general theorems are established that expose the general rules behind certain earlier more concrete results of E. van Douwen [Proc. Amer. Math. Soc. 69 (1978) 183–192] and V.I. Malychin [Proceedings of Leningrad International Topology Conference, Nauka, Leningrad, 1983, pp. 50–61]. In particular, it is proved that every Corson compact space X such that X τ is homogeneous, for some cardinal number τ, is first countable (Corollary 4.9). This shows why every power of the one-point compactification of an uncountable discrete space is not homogeneous [Proc. Amer. Math. Soc. 69 (1978) 183–192]. V.I. Malychin proved that ( ω 1+1) τ is not homogeneous for any τ [Proceedings of Leningrad International Topology Conference, Nauka, Leningrad, 1983, pp. 50–61]. This is covered by Corollary 4.12 below: if some power of a compact scattered space X is homogeneous, then X is countable. It is also shown that, on many occasions, βX is the only compactification of X that can be homogeneous or power homogeneous.