Browsing by Author "Masuret, Jacques"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Closure and compactness in frames
    (Stellenbosch : University of Stellenbosch, 2010-03) Masuret, Jacques; Holgate, D.; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: As an introduction to point-free topology, we will explicitly show the connection between topology and frames (locales) and introduce an abstract notion, which in the point-free setting, can be thought of as a subspace of a topological space. In this setting, we refer to this notion as a sublocale and we will show that there are at least four ways to represent sublocales. By using the language of category theory, we proceed by investigating closure in the point-free setting by way of operators. We de ne what we mean by a coclosure operator in an abstract context and give two seemingly di erent examples of co-closure operators of Frm. These two examples are then proven to be the same. Compactness is one of the most important notions in classical topology and therefore one will nd a great number of results obtained on the subject. We will undertake a study into the interrelationship between three weaker compact notions, i.e. feeble compactness, pseudocompactness and countable compactness. This relationship has been established and is well understood in topology, but (to a degree) the same cannot be said for the point-free setting. We will give the frame interpretation of these weaker compact notions and establish a point-free connection. A potentially promising result will also be mentioned.
  • Thumbnail Image
    Item
    Generalised sequences and compactness notions in point-free topology
    (Stellenbosch : Stellenbosch University, 2017-12) Masuret, Jacques; Holgate, D; Sioen, M; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences.
    ENGLISH ABSTRACT : While sequences and naturally associated notions like convergence and clustering have received extensive attention in classical topology, the same cannot be said for the point-free setting. The aim of this dissertation is to introduce sequences and related sequential notions in frames and to establish the extent to which point-free sequences can characterise countable compactness notions. Furthermore, we will introduce the point-free Dini Property and Strong Dini Property and employ these properties to characterise weaker compactness notions. We also characterise those completely regular frames satisfying the Stone-Weierstrass property.