The general concept of cleavability of a topological space

Abstract

For different classes of spaces P we investigate when a Tychonoff space X is cleavable over P —which means that for every A ⊂ X there exists a continuous mapping f : X → Y ∈ P such that f -1 f( A)= A. In particular, a cleavable space is a Tychonoff space which is cleavable over the class of all separable metrizable spaces. We consider the cases when P consists of all metrizable spaces, or all stratifiable spaces, or all compact Hansdorff spaces, etc.