On zero-dimensional Milutin maps and Michael selection theorems
The authors study the relationship among the following selection theorems of E. Michael: (A) Let \(X\) be a paracompact space with \(\dim X=0\), \(M\) a complete metric space and \(F:X \to M\) a lower-semicontinuous closed- valued map. Then \(F\) admits a single-valued continuous selection. (B) Let \(X\) be a paracompact space, \(B\) a Banach space and \(F:X \to B\) a lower semicontinuous map with closed convex values. Then \(F\) admits a continuous single-valued selection. (C) Let \(X\) be a paracompact space, \(M\) a complete metric space and \(F:X \to M\) a lower semicontinuous map with closed values. Then \(F\) has an upper semicontinuous compact-valued selection, which admits a lower semicontinuous compact-valued selection. It is shown that theorem (A) implies both theorem (B) and (C). This follows from the main result in the paper that every paracompact space is the image of a zero-dimensional paracompact space under a Milutin map. Remark: An alternative proof that theorem (A) implies theorem (C) was given also by \textit{M. Choban} and \textit{E. Michael} in their recent paper [Topology Appl. 49, No. 3, 217-220 (1993; see the review above)].
scientific article; zbMATH DE number 503636
scientific article