Cleavability over reals
A space \(X\) is cleavable over reals \(\mathbb{R}\) if for every subset \(A\) of \(X\) there exists a continuous mapping \(f: X\to\mathbb{R}\) such that \(f^{-1} f(A)= A\). The paper studies the structure of spaces which are cleavable over \(\mathbb{R}\). The main result: Every compactum which is cleavable over \(\mathbb{R}\) is homeomorphic to a subspace of \(\mathbb{R}\); in particular, if a nontrivial continuum is cleavable over \(\mathbb{R}\), then it is homeomorphic to the unit segment \([0,1]\).
scientific article; zbMATH DE number 166417