On the commutativity of the powerspace constructions

TL;DR

This paper studies powerspace constructions on topological spaces, especially quasi-Polish spaces, revealing conditions for their commutativity and how they interact with open set lattices, linked to the property of consonance.

Contribution

It proves that upper and lower powerspaces commute on quasi-Polish spaces and characterizes their distribution over open set lattices, connecting to consonance.

Findings

Upper and lower powerspaces commute on all quasi-Polish spaces.

Commutativity is equivalent to the topological property of consonance.

Complete characterization of powerspace distribution over open set lattices.

Abstract

We investigate powerspace constructions on topological spaces, with a particular focus on the category of quasi-Polish spaces. We show that the upper and lower powerspaces commute on all quasi-Polish spaces, and show more generally that this commutativity is equivalent to the topological property of consonance. We then investigate powerspace constructions on the open set lattices of quasi-Polish spaces, and provide a complete characterization of how the upper and lower powerspaces distribute over the open set lattice construction.