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Bul. Acad. Ştiinţe Repub. Mold. Mat., 2013, Number 2-3, Pages 99–105 (Mi basm338)  

A selection theorem for set-valued maps into normally supercompact spaces

V. Valov

Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7 Canada

Abstract: The following selection theorem is established:
Let $X$ be a compactum possessing a binary normal subbase $\mathcal S$ for its closed subsets. Then every set-valued $\mathcal S$-continuous map $\Phi\colon Z\to X$ with closed $\mathcal S$-convex values, where $Z$ is an arbitrary space, has a continuous single-valued selection. More generally, if $A\subset Z$ is closed and any map from $A$ to $X$ is continuously extendable to a map from $Z$ to $X$, then every selection for $\Phi|A$ can be extended to a selection for $\Phi$.
This theorem implies that if $X$ is a $\kappa$-metrizable (resp., $\kappa$-metrizable and connected) compactum with a normal binary closed subbase $\mathcal S$, then every open $\mathcal S$-convex surjection $f\colon X\to Y$ is a zero-soft (resp., soft) map. Our results provide some generalizations and specifications of Ivanov's results (see [5–7]) concerning superextensions of $\kappa$-metrizable compacta.

Keywords and phrases: continuous selections, Dugundji spaces, $\kappa$-metrizable spaces, spaces with closed binary normal subbase, superextensions.

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Document Type: Article
MSC: 54C65, 54F65
Received: 14.08.2013
Language: English

Citation: V. Valov, “A selection theorem for set-valued maps into normally supercompact spaces”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2013, no. 2-3, 99–105

Citation in format AMSBIB
\Bibitem{Val13}
\by V.~Valov
\paper A selection theorem for set-valued maps into normally supercompact spaces
\jour Bul. Acad. \c Stiin\c te Repub. Mold. Mat.
\yr 2013
\issue 2-3
\pages 99--105
\mathnet{http://mi.mathnet.ru/basm338}


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