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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.).

##### MSC:
 54C65 Selections in general topology