Some properties of cleavable spaces

A topological space X is called cleavable over a class P of spaces if for any A ⊆ X there exists a continuous map f:X → Y such that f(X) = Y ∈P and f−1f(A) = A. It is proved that cleavability over the class of all Tychonoff (metrizable) spaces of weight ⩽ τ is preserved by open perfect maps. The former survives also open compact maps onto normal spaces. Relevant examples are given. A theorem on arcwise connected spaces is proved.