On stable mappings (Q1068389)

This note is a slight generalization of \textit{K. Sakai}'s result for infinite-dimensional fiber bundles [Tsukuba J. Math. 5, 85-99 (1981; Zbl 0483.55011)]. A map \(p: E\to B\) is X-stable if there exists a homeomorphism \(h: E\times X\to E\) such that p h\(=p \pi_ E\) \((\pi_ E: E\times X\to E\) is the projection). p is called locally X-stable if for every point \(x\in E\) there exists a neighborhood U of x such that \(p| U\) in X-stable. By comparing the obtained result with Sakai's result, the author replaces the local triviality assumption by the locally X-stable assumption. No applications apart from already known results are given.