Some properties of cleavable spaces (Q1346170)

A space \(X\) is said to be cleavable over a class \({\mathcal P}\) of topological spaces if for every subset \(A\) of \(X\) there are \(Y\in {\mathcal P}\) and a continuous mapping \(f: X\to Y\) such that \(f^{-1} f(A)=A\). The author proves that cleavability over the class of all spaces having weight \(\leq\tau\) and over the class of all metrizable spaces (with weight \(\leq\tau\)) is an invariant of open perfect mappings. If an arcwise connected space is cleavable over the class \({\mathcal P}\) of all spaces of countable weight, then it admits a continuous bijection onto a space in \({\mathcal P}\). Two examples are given.