On the cardinality of power homogeneous compacta (Q1763608)

A space \(X\) is power homogeneous iff some \(X^{\kappa}\) is homogeneous. This turns out to be a fairly strong property, which restricts cardinal invariants and their relationships. This paper studies compact Hausdorff power homogeneous spaces. The main theorem is: if \(X\) is compact Hausdorff power homogeneous then w\((X)^{\pi_{\chi}(X)} \geq | X| \). The main corollary is that, for such \(X\), \(| X| \leq 2^{c(X) \cdot \pi_{\chi}(X)}\). There are a number of other results, e.g., for such spaces, \(| X| \leq 2^{s(X)}\) and \(2^{\chi(X)} \leq 2^{s(X)}\). Several examples and counterexamples are given.