On power homogeneous spaces and related problems
by
Alexander Arhangelskii
Ohio University

A T₁-space X is said to be power homogeneous if, for some non-zero cardinal number \tau, the space X^\tau is homogeneous.

Let \tau be an infinite cardinal number. Recall that the character of a space X at a point x does not exceed \tau if there exists a base B_x at x such that |B_x| <= \tau.

We are going to prove the following theorem:

If X is a power homogeneous compact Hausdorff space then, for every infinite cardinal number \tau, the set of all points of X, at which the character of X is not greater than \tau, is closed in X.

It is easy to establish with the help of this theorem that many well known compact spaces are not power homogeneous. The theorem will follow from a considerably more general statement which will be established after a series of preliminary results. A part of the technique is a binary operation of a very general nature which can be introduced on many topological spaces.

Date received: June 3, 2003