On the Problem of Condensation onto Compact Spaces

Abstract

Assuming the continuum hypothesis CH, it is proved that there exists a perfectly normal compact topological space Z and a countable set $$E \subset Z$$ such that $$Z{\backslash }E$$ is not condensed onto a compact space. The existence of such a space answers (in CH) negatively to V.I. Ponomarev’s question as to whether every perfectly normal compact space is an $$\alpha $$-space. It is proved that, in the class of ordered compact spaces, the property of being an $$\alpha $$-space is not multiplicative.