Selections of maps with nonclosed values and topologically regular maps (Q1890809)

The paper begins by defining when a map \(p : E \to B\), with metric domain \(E\) and homeomorphic fibers, is topologically regular. The principal result is: Theorem 1. Let \(p : E \to B\) be a topologically regular map, with \(E\) and \(B\) compact metric (slightly weaker assumptions on \(E\) and \(B\) suffice) and with all fibers homeomorphic to \([0,1]\). Then \(p\) is a locally trivial fibration. Reviewer's comment: This theorem was also proved, by similar methods, by \textit{C. P. Pixley} [Ill. J. Math. 20, 22-29 (1976; Zbl 0328.54012); Theorem W]. As Pixley noted, the theorem had previously been obtained, by quite different methods, by \textit{H. Whitney} [Ann. Math., II. Ser. 34, 244-270 (1933; Zbl 0006.37101); Theorem 14A].