Controlling Tietze-Urysohn extensions
Let \(A\) be a closed subset of the normal Hausdorff space \(X\), and let \(f\) be a continuous real-valued function with domain \(A\) (denoted \(f\in C(A)\)). The author investigates when \(f\) can be extended to an \(f^*\in C(X)\) that satisfies various sorts of pre-assigned conditions. The following results are representative but not exhaustive: Theorem. If \(f\in C(A)\) such that \(f[A]= [c,d ]\), if \(f^\leftarrow (c)\neq \emptyset \neq f^\leftarrow (d)\), and if \(H\) and \(K\) are disjoint closed \(G_\delta\)-sets of \(X\) for which \(H\cap A= f^\leftarrow (c)\) and \(K\cap A= f^\leftarrow (d)\), then there exists \(f^*\in C(X)\) for which \(f^*|A=f\), \((f^*)^\leftarrow (c)= H\), and \((f^*)^\leftarrow (d)= K\). Theorem. Let \(A\) be closed in the compact metric space \(X\), let \(f,g\in C(A)\), and let \(h\in C(X)\) with \(fg= h|A\). If \(g\geq 0\) and \(g^\leftarrow (0) \subseteq f^\leftarrow (0)\), then there exist \(f^*, g^*\in C(X)\) with \(f^*|A=f\), \(g^*|A=g\) and \(f^* g^* =h\).
scientific article; zbMATH DE number 815788