Some problems about co-consonance of topological spaces

TL;DR

This paper investigates the properties of co-consonance in topological spaces and their powerspaces, establishing conditions under which co-consonance is preserved or implied across these structures.

Contribution

It proves that retracts of (co-)consonant spaces are (co-)consonant and explores the relationships between co-consonance of spaces and their powerspaces under various conditions.

Findings

Retracts of (co-)consonant spaces are (co-)consonant.

Co-consonance of Smyth powerspace implies co-consonance of the original space under certain conditions.

Co-consonance of the lower powerspace implies co-consonance of the space.

Abstract

In this paper, we first prove that the retract of a consonant space (or co-consonant space) is consonant (co-consonant). Using this result, some related results have obtained. Simultaneously, we proved that (1) the co-consonance of the Smyth powerspace implies the co-consonance of a topological space under a necessary condition; (2) the co-consonance of a topological implies the co-consonance of the smyth powerspace under some conditions; (3) if the lower powerspace is co-consonant, then the topological space is co-consonant; (4) the co-consonance of implies the co-consonance of the lower powerspace with some sufficient conditions.