Chapter 17 - Selection problems for hyperspaces

Abstract

This chapter discusses selection problems for hyperspaces. For a T1-space X, let ℱ(X) be the set of all nonempty closed subsets of X. ℱ(X) is endowed with the Vietoris topology TV, and called the Vietoris hyperspace of X. A space X is orderable (or linearly orderable) if the topology of X coincides with the open interval topology on X generated by a linear ordering on X. A space X is sub-orderable (or generalized ordered) if it can be embedded into an orderable space. A space X is weakly orderable if there exists a coarser orderable topology on X. In all these cases, the corresponding linear order on X is called compatible for the topology of X or a compatible order for X. A selection f : ℱ2(X) →X is usually called a weak selection for X. This chapter elaborates about weak selections and properties that follow from orderability, and it discusses the concepts of topological well-ordering and selections, in detail. A discussion on selections and disconnectedness-like properties is also presented in the chapter.