On the cardinality of power homogeneous compacta

We prove that if X is a power homogeneous compact space then |X|⩽2c(X)·πχ(X). This generalizes similar results of Arhangelʹskiı̆, van Douwen and Ismail. We apply this result to get new estimates for the cardinality of (power) homogeneous compacta satisfying some special conditions.