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.
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