Unicoherence in Locales

TL;DR

This paper extends the concept of unicoherence, a topological connectedness property, to locales, exploring its properties and showing that many classical characterizations carry over, including those involving separation properties.

Contribution

It introduces and investigates the notion of unicoherence within the framework of locales, generalizing classical topological results.

Findings

Many classical characterizations of unicoherence extend to locales

Separation properties are involved in the locale setting

The paper provides new insights into locale connectedness

Abstract

In this paper, we generalize the concept of unicoherence to the context of frames. Unicoherence, originally introduced by Kuratowski, is a connectedness property that is well studied in classical topology and used to detect holes of a space. We extend the notion of unicoherence to locales and we then investigate its properties. In particular, we prove that many of the known characterizations of unicoherence for topological spaces extend to the setting of locales. Some of these characterizations interestingly involve separation properties for locales.