Selections and suborderability

Abstract

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 or first countable, or connected. An infinite pseudocompact topological group with such a continuous selection is homeomorphic to the Cantor set. 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, \eg\ a locally compact locally countable monotonically normal space with a continuous zero-selection which is not suborderable