Vietoris topology on hyperspaces associated to a noncommutative compact space

TL;DR

This paper explores the topology of hyperspaces related to noncommutative compact spaces, extending classical concepts like Vietoris topology and distances to the quantum setting, and investigates their properties and open problems.

Contribution

It introduces a Vietoris-type topology on hyperspaces of noncommutative compact spaces and studies metric properties in the quantum metric space framework.

Findings

Defined a Vietoris topology for hyperspaces of noncommutative spaces

Analyzed Hausdorff and infimum distances in quantum metric spaces

Formulated open problems on distances between sub-circles of quantum tori

Abstract

We study some topological spaces that can be considered as hyperspaces associated to noncommutative spaces. More precisely, for a NC compact space associated to a unital C*-algebra, we consider the set of closed projections of the second dual of the C*-algebra as the hyperspace of closed subsets of the NC space. We endow this hyperspace with an analog of Vietoris topology. In the case that the NC space has a quantum metric space structure in the sense of Rieffel we study the analogs of Hausdorff and infimum distances on the hyperspace. We also formulate some interesting problems about distances between sub-circles of a quantum torus.