A characterization of power homogeneity (Q2469568)

Let \(X\) be a T\(_2\) space. \(X\) is power homogeneous if \(X^\kappa\) is homogeneous for some cardinal \(\kappa\), \(X^A\) is \(\Delta\)-homogeneous if any two points on the small diagonal of \(X^A\) can be mapped onto each other by a homeomorphism of \(X^A\) and \(X\) is \(\Delta\)-power homogeneous if \(X^\kappa\) is \(\Delta\)-homogeneous for some \(\kappa\). Under mild conditions on \(\kappa\) the following four conditions are equivalent: \(X^\kappa\) is homogeneous; \(X\) is power homogeneous; \(X\) is \(\Delta\)-power homogeneous; \(X^{\pi w(X)}\) is \(\Delta\)-homogeneous.