Characterizations of Ordinals via Selectors by U. Marconi, J. Pelant and L. Rotter

Characterizations of Ordinals via Selectors

U. Marconi, J. Pelant and L. Rotter

Research Announcement

In 1997 S. Fujii and T. Nogura proved that a compact Hausdorff space is homeomorphic to a compact ordinal iff there exists a Vietoris continuous selector which associates to every closed set F an isolated point of F. By using the Fell topology, we show that all ordinal spaces can be characterized via continuous selectors.

Theorem 1. The following conditions are equivalent for an Hausdorff space X:

X is homeomorphic to an ordinal space;
there exists a selector continuous for the Fell topology which associates to every closed set F an isolated point ofF.

It is still an open problem characterizing ordinal spaces via Vietoris-continuous selectors. A partial answer is given by the following theorem, which characterizes a larger class of ordinals, including all ordinals with uncountable cofinality.

Theorem 2. Let X be a normal space with a unique compactification. The following are equivalent:

X is homeomorphic to an ordinal space;
there exists a Vietoris continuous selector which associates to every closed set F an isolated point of F.

These results were presented at the ``Second Italian-Spanish Conference on General Topology and its Applications" (Trieste, September 1999).

The Fell topology has been used independently by Gutev and Nogura to obtain other results on orderability of spaces.

Date Received: September 24, 1999