Condensations of Cartesian Products
by
Oleg Pavlov
Ohio University

All considered spaces are Tychonoff; condensation is a one-to-one continuous mapping.

Theorem 1. If X is not a pseudocompact space and |X| is a non-measurable cardinal, then some power of X can be condensed onto a \sigma-compact space.

Therefore, if X is not pseudocompact in some power and |X| is not measurable, then some larger power of X can be condensed onto a \sigma-compact space. Theorem 1 is not valid for any Tychonoff space of non-measurable cardinality.

Theorem 2. Let X be countably compact in every power. Then there exists a larger space M(X) such that for any cardinal \mu and any condensation f, f((M(X))^\mu) contains a closed subset homeomorphic to X^\mu.

In particular, (M(X))^\mu can not be condensed on a \sigma-compact space if X is not compact.

Date received: March 16, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caam-45.