Triquotient and inductively perfect maps
A map \(f: X\to Y\) of topological spaces is said to be triquotient if for each \(y\in Y\) there is a family \(\eta_ y\) of open subsets of X such that \(X\in \eta_ y\), and the following conditions are satisfied: (i) if \(U\in \eta_ y\) and \(\gamma\) is an open cover of \(U\cap f^{-1}(y)\) by open subsets of X, then there exist a finite number of elements \(U_ i\in \gamma\), \(1\leq i\leq k\), such that \(\cup^{k}_{i=1}U_ i\in \eta_ y\); (ii) if \(U\in \eta_ y\), then \(y\in int f(u)\); (iii) if \(U\in \eta_ y\), then there is a neighborhood 0(y) such that \(U\in \eta_{\xi}\) for every \(\xi\in 0(y)\). Triquotient maps were introduced by \textit{E. Michael} [Ill. J. Math. 21, 716-733 (1977; Zbl 0386.54007)]. A mapping \(f: X\to Y\) of topological spaces is said to be inductively perfect if there is a closed set \(F\subset X\) such that \(f(F)=Y\) and f/F is perfect. The main results of this paper are as follows: Theorem 1. Let f:X\(\to Y\) be a triquotient map onto a paracompact space Y. If there is a perfect extension \(f^*: X^*\to Y\) of f such that X is a \(G_{\delta}\) set in \(X^*\), then f is inductively perfect. Theorem 2. If f:X\(\to Y\) is a triquotient map of a metric space (X,d) onto a metric space Y such that every fiber \(f^{-1}(y)\) is a complete metric space with respect to d and for every open \(U\subset X\), f(U) is an \(\epsilon_{\delta}\)-set in Y, then f is inductively perfect.
scientific article; zbMATH DE number 3949272
scientific article