Cleavable topological spaces
A topological space \(X\) is said to be cleavable over a class \({\mathcal P}\) of spaces if for every \(A \subset X\) there exist \(Y \in {\mathcal P}\) and a continuous mapping \(f : X \to Y\) such that \(f^{-1} f(A) = A\). The authors consider the following general question: if \(X\) is cleavable over a class \({\mathcal P}\) and there is a continuous mapping from \(X\) onto a space in \({\mathcal P}\), does \(X\) admit a continuous bijection (a homeomorphism) onto a space in \({\mathcal P} \)? A typical result is: Let \(\tau\) be an infinite cardinal number. If a space \(X\) is cleavable over a \(\tau\)-multiplicative and hereditary class \({\mathcal P}\) and there is a continuous mapping from \(X\) onto a space \(Y \in {\mathcal P}\) such that \(L(f^{-1} (y)) \leq \tau\) for every \(y \in Y\), then \(X\) admits a continuous bijection onto a space \(Z \in {\mathcal P}\) in every of the following cases: (1) \(Y \in {\mathcal P}\) and \(L(Y) \leq \tau\) imply \(\psi (Y) \leq \tau\) and \(|Y |\leq 2^\tau\) (Theorem 5); (2) for every \(Y \in {\mathcal P}\), \(\psi (Y) \leq \tau\) and \(t(Y) \leq \tau\) (Theorem 6). Some corollaries of these general results are derived. The assertion ``if a space \(X\) is cleavable over the class of Corson (Eberlein) spaces and there is a perfect mapping of \(X\) onto a Corson (Eberlein) space, then \(X\) is also a Corson (Eberlein) space'' is consistent with ZFC. The paper contains no proofs.
scientific article; zbMATH DE number 848897