We extend van Mill-Wattel's results and show that each countably compact completely regular space with a continuous selection on couples is suborderable. The result extends also to pseudocompact spaces if they are either scattered, first countable, or connected. An infinite pseudocompact topological group with such a continuous selection is homeomorphic to the Cantor set. A zero-selection is a selection on the hyperspace of closed sets which chooses always an isolated point of a set. Extending Fujii-Nogura results, we show that an almost compact space with a continuous zero-selection is homeomorphic to some ordinal space, and a (locally compact) pseudocompact space with a continuous zero-selection is an (open) subspace of some space of ordinals. Under the Diamond Principle, we construct several counterexamples, e.g. a locally compact locally countable monotonically normal space with a continuous zero-selection which is not suborderable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-1,
author = {Giuliano Artico and Umberto Marconi and Jan Pelant and Luca Rotter and Mikhail Tkachenko},
title = {Selections and suborderability},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {1-33},
zbl = {1019.54014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-1}
}
Giuliano Artico; Umberto Marconi; Jan Pelant; Luca Rotter; Mikhail Tkachenko. Selections and suborderability. Fundamenta Mathematicae, Tome 173 (2002) pp. 1-33. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-1/