Ordinals and set-valued zero-selections for hyperspaces

A continuous zero-selection f for the Vietoris hyperspace F ( X ) of the nonempty closed subsets of a space X is a Vietoris continuous map f : F ( X ) → X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F ( X ) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection.