A theorem on homotopy lifting and characterization of certain soft maps (Q1175345)

Recall the notion of a soft (resp. locally soft) map \(f:X\to Y\) [\textit{E. V. Shchepin} Russ. Math. Surv. 39, No. 5, 251-270 (1984); translation from Usp. Mat. Nauk 39, No. 5(239), 209-224 (1984; Zbl 0568.57012)]: \(f:X\to Y\) is soft (resp. locally soft) with respect to a pair \((A,B)\) of a topological space \(A\) and its subspace \(B\) provided that for any map \(\varphi:A\to Y\) and any lift \(\alpha:B\to X\) of \(\varphi| B\) there is a lift \(\varphi':A\to X\) [resp. \(\varphi':U\to X\), \(U\) a neighborhood of \(B\) in \(A]\) of \(\varphi\) such that \(\varphi'| B=\alpha\). \(f:X\to Y\) is soft (resp. locally soft) if it is soft (resp. locally soft) with respect to any pair \((A,B)\) consisting of a paracompact space \(A\) and its closed subspace \(B\). The author proves the following analogue of the homotopy extension theorem: Any locally soft map \(f:X\to Y\) is soft with respect to the pair \((A\times[0,1]\), \(A\times\{0,1\}\cup B\times[0,1])\). Also, a sufficient condition for a map to be (locally) soft is given.