On power homogeneous spaces (Q1612232)

A topological space \(X\) is called power homogeneous if there exists a cardinal number \(\tau>0\) such that the space \(X^\tau\) is homogeneous. It may easily happen that a space \(X\) is not homogeneous while \(X\) is power homogeneous. Every first countable zero-dimensional Hausdorff space is \(\omega\)-power homogeneous. In this paper, some new necessary conditions for a space to be power homogeneous are obtained. The notions of a Moscow space and a weakly Klebanov space are applied to study power homogeneous spaces. In particular it is proved that (1) every Corson compact power homogeneous space is first countable; (2) a compact scattered space is power homogeneous if and only if it is countable.