Fiber properties of closed maps, and weak topology (Q2701861)

The authors investigate the following question:NEWLINENEWLINENEWLINELet \(f:X\to Y\) be a closed map. Under what conditions on \(X\) or \(Y\) does \(Bf^{-1}(y)\) have some nice properties for each \(y\in Y\)?NEWLINENEWLINENEWLINELet denote \({\mathcal P}^*\) a point-countable closed cover determining a space and \({\mathcal F}\) a closed cover dominating a space. The authors obtain many results, one of them is the following NEWLINENEWLINENEWLINETheorem: Let \(f:X\to Y\) be a closed map such that \(X\) is an \(S_\alpha\)-space. Then the following (A) and (B) hold:NEWLINENEWLINENEWLINE(A) Let \(X\) be a bi-\(k\)-space. If every closed paracompact \(M\)-subspace of \(Y\) is locally \(\alpha\)-compact, then each \(Bf^{-1}(y)\) is locally \(\alpha\)-compact.NEWLINENEWLINENEWLINE(B) Let \(Y\) have a cover \({\mathcal F}\) (or \({\mathcal P}^*)= \{Y_\gamma\}\), where each \(Y_\gamma\) is one of the following spaces: (i) an inner-closed \(A\)-space, (ii) a locally \(\alpha\)-compact space, (iii) a sequential space which contains no closed copy of \(S_\alpha\). Then (a) and (b) below hold: (a) Suppose that \(X\) is a bi-quasi \(k\)-space or an inner-closed \(A\)-space with \(t(X)\leq \omega\). Then each \(Bf^{-1} (y)\) is a locally \(\alpha\)-compact space dominated by a closed cover \({\mathcal C}\) of \(\alpha\)-compact subsets (where \(\alpha= \omega\) if all \(Y_\gamma\) are (i)). For \({\mathcal P}^*\) the closed cover \({\mathcal C}\) can be countable. (b) Suppose that \(X\) is a singly bi-quasi \(k\)-space. Then the set \(\{y\in Y:Bf^{-1}(y)\) is not \(\alpha\)-compact\} is closed discrete in \(Y\) For \({\mathcal P}^*\) each \(Bf^{-1}(y)\) is a countable union of \(\alpha\)-compact closed subsets.