Bul. Acad. Ştiinţe Repub. Mold. Mat., 2013, Number 2-3, Pages 99–105
A selection theorem for set-valued maps into normally supercompact spaces
Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7 Canada
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.
PDF file (143 kB)
MSC: 54C65, 54F65
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
\paper A selection theorem for set-valued maps into normally supercompact spaces
\jour Bul. Acad. \c Stiin\c te Repub. Mold. Mat.
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