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A simultaneous selection theorem. (English) Zbl 1257.54025

As a generalization of Tietze’s extension theorem, E. Michael’s selection theorem [Ann.Math.(2) 63, 361–382 (1956; Zbl 0071.15902)] and J. Dugundji’s simultaneous extension theorem [Pac.J. Math.1, 353-367 (1951; Zbl 0043.38105)] are well-known. In this paper, the author generalizes in a way both of the above two theorems by proving a simultaneous selection theorem.
By a space, we mean a completely regular Hausdorff space. For a space \(X\) and a linear topological space \(E\), let \(C(X,E)\) denote the linear space of all continuous mappings from \(X\) to \(E\). Let \(2^Y\) denote the power set of a space \(Y\). A mapping \(\Phi : X \to 2^Y\) is said to be lower semicontinuous if for every open subset \(V\) of \(Y\), the set \(\{ x \in X: \Phi (x) \cap V \neq \emptyset\}\) is open in \(X\). The author establishes the following theorem: Let \(X\) be a paracompact \(k\)-space, \(Y\) a completely metrizable space, \(E\) a locally convex complete linear topological space and \(\Phi : X \to 2^Y\setminus\{\emptyset\}\) a lower semicontinuous mapping. Then there exists a linear operator \(S: C(Y, E) \to C(X,E)\) such that \(S(f)(x)\) is contained in the closed convex hull of \(f(\Phi(x))\) for every \(x \in X\) and \(f \in C(Y,E)\). Furthermore, \(S\) is continuous with respect to the topology of uniform convergence on compact subsets.
Applying this theorem, the author also proves a generalization of A. A. Milyutin’s theorem [Teor.Funkts., Funkts.Anal.Prilozh.2, 150–156 (1966; Zbl 0253.46050)].

MSC:

54C65 Selections in general topology
54C20 Extension of maps
46B03 Isomorphic theory (including renorming) of Banach spaces
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