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Unified selection and factorization theorems. (English) Zbl 0622.54015
Selection and factorization theorems are established for the members of two new classes of set-valued mappings. Thus some results of E. Michael [Pac. J. Math. 87, 189-197 (1980; Zbl 0405.54016)], S. I. Nedev, Serdica 6, 291-317 (1980; Zbl 0492.54006)], and F. S. De Blasi and J. Myjak [Proc. Am. Math. Soc. 93, 369-372 (1985; Zbl 0565.54013)] are generalized and obtained from a common point of view. For instance: The set-valued mapping \(\phi\) : \(X\to 2^ Y\) is called strongly-selection-factorization semicontinuous (s.s.f.s.c.) if for every \(\epsilon >0\), for every open locally finite covering \(\gamma\) of X with \(Card(\gamma)\leq w(Y)\cdot \aleph_ 0\) and every collection \(\{\) y(x)\(\subset \phi (x):\) \(0<Card(y(x))<\aleph_ 0\) and \(x\in X\}\) there exists an open locally finite covering \(\omega\) of X with \(Card(\omega)\leq w(Y)\cdot \aleph_ 0\) and a mapping \(r: \omega\to \gamma\) such that for every U in \(\omega\), \(U\subset r(U)\) and there is a point \(x_ u\) of r(U) for which \(y(x_ u)\subset \cap \{B_{\epsilon}(\phi (x)):\) \(x\in U\}\). Here (Y,d) is a metric space and \(B_{\epsilon}(A)=\{y\in Y:\) \(d(y,a)<\epsilon\) for some \(a\in A\}\) for \(A\subset Y\). Let P be a class of set-valued mappings. For a particular mapping \(\phi\) we say that \(\phi\) has the SEP(P) if \(\phi\in P\) and whenever \(A\subset X\), is closed, every single-valued continuous selection g for \(\phi| A\) such that \(\phi_ g\in P\) (where \(\phi_ g(x)=g(x)\) for \(x\in A\) and \(\phi_ g(x)=\phi (x)\) for \(x\in X\setminus A)\) extends to a single-valued continuous selection for \(\phi\). If g only extends to a single-valued continuous selection for \(\phi| U\) for some neighbourhood U of A in X, then we say that \(\Phi\) has the SNEP(P). Theorem. Let X be a normal space, Y a Banach space, \(Z\subset X\) with \(\dim_ XZ\leq n+1\), and \(\phi\) :X\(\to {\mathfrak F}(Y)\) be l.s.c. and s.s.f.s.c. simultaneously, with \(\phi\) (x) convex for all \(x\in X\setminus Z\) and with \(\{\phi\) (x): \(x\in Z\}\) uniformly \(equi\)-LC\({}^ n\). Then \(\phi\) has the SNEP(s.s.f.s.c.). If, moreover, \(\phi\) (x) is \(C^ n\) for every \(x\in Z\), then \(\phi\) has the SEP(s.s.f.s.c.).

54C65 Selections in general topology