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Unicoherence in Locales

Elena Caviglia, Luca Mesiti, Cerene Rathilal

TL;DR

This work generalizes the classical notion of unicoherence from topological spaces to locales by formulating unicoherence in the pointfree setting of frames and sublocales. It develops the notions of continua and regions within locales, extends known topological characterizations to locale theory, and establishes a robust suite of equivalent conditions linking unicoherence to boundary behavior, separation properties, and open/unicoherence concepts. The results encompass a broad network of equivalences, including Brouwer-type properties and Phragmen–Brouwer-type conditions, and illustrate the theory with examples such as $\mathcal{O}X$ reflecting the properties of $X$. Overall, the paper advances pointfree topology by providing a cohesive, technically rich treatment of unicoherence for locales, with constructive proofs and new tools for analyzing connectedness through sublocales, closures, interiors, and boundaries.

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.

Unicoherence in Locales

TL;DR

This work generalizes the classical notion of unicoherence from topological spaces to locales by formulating unicoherence in the pointfree setting of frames and sublocales. It develops the notions of continua and regions within locales, extends known topological characterizations to locale theory, and establishes a robust suite of equivalent conditions linking unicoherence to boundary behavior, separation properties, and open/unicoherence concepts. The results encompass a broad network of equivalences, including Brouwer-type properties and Phragmen–Brouwer-type conditions, and illustrate the theory with examples such as reflecting the properties of . Overall, the paper advances pointfree topology by providing a cohesive, technically rich treatment of unicoherence for locales, with constructive proofs and new tools for analyzing connectedness through sublocales, closures, interiors, and boundaries.

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.

Paper Structure

This paper contains 5 sections, 29 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Properties of Sublocales
  3. On Connectedness
  4. Unicoherent Locales
  5. Main Results

Key Result

Proposition 2.2

Let $S\subseteq L$ be a sublocale. A closed sublocale $\mathpzc{c}\xspace_{S}(u)$ of $S$ is the same as $\mathpzc{c}\xspace_{S}(u)\cap S$. An open sublocale $\mathpzc{o}\xspace_{S}(u)$ of $S$ is the same as $\mathpzc{o}\xspace_{}(u)\cap S$. Let $T\subseteq S$ be a sublocale. Then the closure $\overl

Theorems & Definitions (69)

  • Definition 2.1: topwop
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 59 more