Continuous selections for weakly Hausdorff lower semicontinuous multifunctions.

*(English)*Zbl 0565.54013Let us denote the set of all nonempty closed convex sets of the Banach space Y by \(C_ c(Y)\), and let us say that the multifunction F from a topological space X to \(C_ c(Y)\) is weakly Hausdorff lower semicontinuous \((H_ w-l.s.c.)\) if for all \(x_ 0\in X\), for all \(\epsilon >0\), and for all V neighbourhood of \(x_ 0\) there exists a neighbourhood U of \(x_ 0\), \(U\subset V\), and there exists x’\(\in U\) such that for all \(x\in U\) we obtain \(F(x')\subset F(x)+\epsilon S\), where S is the unit sphere in Y. A geometrical lemma enables us to use a result of Deutsch and Kenderov to deduce the following result: Let X be paracompact. Then every \(H_ w-l.s.c\). multifunction \(F: X\to C_ c(Y)\) admits a continuous selection. This theorem is similar to Michael’s selection theorem but does not follow from it.

Reviewer: P.Holicky

##### Keywords:

continuous selection; convex closed set; paracompact space; Banach space; multifunction; weakly Hausdorff lower semicontinuous
PDF
BibTeX
XML
Cite

\textit{F. S. De Blasi} and \textit{J. Myjak}, Proc. Am. Math. Soc. 93, 369--372 (1985; Zbl 0565.54013)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Francesco S. De Blasi and Józef Myjak, Sur l’existence de sélections continues, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 17, 737 – 739 (French, with English summary). · Zbl 0527.54016 |

[2] | Frank Deutsch and Petar Kenderov, Continuous selections and approximate selection for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14 (1983), no. 1, 185 – 194. · Zbl 0518.41031 |

[3] | M. Edelstein and A. C. Thompson, Some results on nearest points and support properties of convex sets in \?\(_{0}\), Pacific J. Math. 40 (1972), 553 – 560. · Zbl 0202.39503 |

[4] | Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361 – 382. · Zbl 0071.15902 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.