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
Аннотация:
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.
Ключевые слова и фразы:
continuous selections, Dugundji spaces,
$\kappa$-metrizable spaces, spaces with closed binary normal subbase, superextensions.
MSC: 54C65,
54F65 Поступила в редакцию: 14.08.2013
Язык публикации: английский