Set-valued maps and \(AE(0)\)-spaces
For a topological space \(X\), \(C(X)\) will denote the set of all real-valued (continuous) maps on \(X\). If \(X \subset Y\), then \(C(Y) |X\) is the set of all elements of \(C(X)\) extendable to the whole of \(Y\). Given a map \(f : X \to Y\), \(C(f) : C(Y) \to C (X)\) denotes the operator induced by \(f\). A space \(X\) is an \(AE (n)\)-space \((n = 0, 1, \dots)\) if for any space \(Z\) with \(\dim Z \leq n\) and for any subspace \(Z_0 \subset Z\), each map \(f : Z_0 \to X\) with \(C(f) (C(X)) \subset C(Z) |Z_0\) can be extended over \(Z\). The authors characterize non-compact \(AE (0)\)- and \(AE (1)\)-spaces in terms of set-valued maps. The main result of the paper states that a completely regular space \(X\) is an \(AR (0)\) if and only if any set-valued map \(F : Y \to X\) from this class, where \(\dim Y = 0\), has a continuous selection.
scientific article; zbMATH DE number 500696
scientific article