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On many-valued continuous selections. (English. Russian original) Zbl 0562.54026

Russ. Math. Surv. 39, No. 3, 213-214 (1984); translation from Usp. Mat. Nauk 39, No. 3(237), 241-242 (1984).
Let X be a paracompact and Y a metric space. \({\mathcal F}(Y)\) denotes the collection of all nonempty closed subsets of Y, \(S(Y)=\{F\in {\mathcal F}(Y):\) F is a compact\(\}\), \(C(Y)=\{F\in {\mathcal F}(Y):\) F is a continuum\(\}\). Developing the theorem of E. Michael [Duke Math. J. 26, 647-651 (1959; Zbl 0151.308)] the author formulates the following main result. Let \(\phi\) : \(X\to {\mathcal F}(Y)\) be lower semicontinuous with complete images \(\phi\) (x). The family \(\{\) \(\phi\) (x): \(x\in X\}\) be uniformly locally connected (a family \({\mathcal S}\subset {\mathcal F}(Y)\) is called u.l.c. if for any \(y\in \cup {\mathcal S}\) and any neighbourhood U of it (in Y) there exists a neighbourhood V (in Y) such that every pair a,b\(\in V\cap S\), \(S\in {\mathcal S}\) can be joined in \(U\cap S\) by a connected set). Then any continuous multivalued selection \(\psi\) : \(A\to S(Y)\) [A\(\to C(Y)]\) of \(\phi\) where \(A=\bar A\subset X\) (i.e. \(\psi\) (x)\(\subset \phi (x)\), for all \(x\in A)\) can be extended to a continuous multivalued selection \({\tilde \psi}\): \(U\to S(Y)\) [respectively, \({\tilde \psi}\): \(U\to C(Y)]\) where U is a neighbourhood of A. If, in addition, all \(\phi\) (x) are connected U may be equal to X. Some corollaries are considered, including a generalization of the classic Michael’s continuous selection theorem.
Reviewer: V.V.Obuhovskii

MSC:

54C65 Selections in general topology

Citations:

Zbl 0151.308
Full Text: DOI