Orderability and continuous selections for Wijsman and Vietoris hyperspaces

Abstract

Bertacchi and Costantini obtained some conditions equivalent to the existence of continuous selections for the Wijsman hyperspace of ultrametric Polish spaces. We introduce a new class of hypertopologies, the macro-topologies. Both the Wijsman topology and the Vietoris topology belong to this class. We show that subject to natural conditions, the base space admits a closed order such that the minimum map is a continuous selection for every macro-topology. In the setting of Polish spaces, these conditions are substantially weaker than the ones given by Bertacchi and Costantini. In particular, we conclude that Polish spaces satisfying these conditions can be endowed with a compatible order and that the minimum function is a continuous selection for the Wijsman topology, just as it is for [0; 1]. This also solves a problem implicitely raised in Bertacchi and Costantini's paper.

Keywords

• Selection
• Vietoris topology
• Wijsman topology
• macro-topology
• ∆-topology
• Ordered space
• Compatible order
• Sub-compatible order
• Extra-dense set
• Lexor
• Complete lexor
• Polish space
• Star-set
• n-coordinated-function
• n-coordinated-set