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The abstract of the talk (see below) may be downloaded in one of the following formats:
\def\FX#1{\ensuremath{\mathfrak{F}_{#1}(X)}}
We study continuous selectors $\sigma :\FX\tau \to X$ where
$\FX\tau$ is a hyperspace of non-empty closed subsets of $X$
equipped with a topology $\tau$ and $\sigma (F)\in F$ for each
$F\in \FX{}$. This topic has been investigated for years and we
are going to mention some new results and to relate them to other
results in the literature.
Some sample facts from our background:
\begin{thm}[Mazurkiewicz-Sierpi\'nski]\
$X$ is first countable scattered compact $\iff$ $X$ is
homeomorphic to a countable ordinal $\iff$ $X$ is compact and
countable.
\end{thm}
A zero-selector on $X$ is a selector $\sigma$ such that
$\sigma(F)$ is isolated relatively to~$F$.
If $X$ has a zero-selector, then $X$ is scattered. Moreover, any
subspace of ordinals has a zero-selector--just take minima of
non-empty closed sets.
A first-countable paracompact scattered space is a completely
metrizable subspace of an ordinal space (Telg\' arsky);
consequently it has a zero-selector.
\begin{thm}[Fujii-Nogura]
If $X$ is compact and there exist a zero-selector, then $X$ is
homeomorphic to an ordinal space.
\end{thm}
New and also older results mentioned in this lecture were obtained
jointly with G.Artico, U.Marconi, L.Rotter and M.Tkachenko.
\begin{thebibliography}{10}
\bibitem{AM99}
G.~Artico and U.~Marconi, {\em Selections and topologically
well-ordered
spaces}, Topology Appl., to appear (2000), 6 pages.
\bibitem{AMPRT99}G.~Artico,
U.~Marconi, J.~Pelant, L.~Rotter, M.~Tkachenko,{\em Selections and
suborderability}, submitted.
\bibitem{GutNog99}
V.~Gutev and T.~Nogura, {\em Fell continuous selections and
topologically
well-orderable spaces}, Preprint (1999), 8 pages.
\bibitem{Nog97}
S.~Fujii and T.~Nogura, {\em Characterizations of compact ordinal
spaces via
continuous selections}, Topology Appl., {\bf 20} (1997), 1--5.
\bibitem{MiWat81}
J.~van Mill and E.~Wattel, {\em Selections and
orderability},
Proc. Amer. Math. Soc. {\bf 83} (1981), 601--605.
\bibitem{MiWat84}
J.~van Mill and E.~Wattel, {\em Orderability from selections:
Another solution to the
orderability problem}, Fund. Math. {\bf 121} (1984), 219--229.
\bibitem{Pur85}
S.~Purish, {\em Scattered compactifications and the orderability
of scattered
spaces, {II}}, Proc. Amer. Math. Soc. {\bf 95} (1985), 636--640.
\bibitem{Tel68}
R.~Telg\'{a}rsky, {\em Total paracompactness and paracompact
dispersed spaces},
Bull. Polish Acad. Sci. Math. {\bf 16} (1968), 567--572.
\end{thebibliography}