Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / 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. / AFRIKAANSE OPSOMMING: As 'n inleiding tot punt-vrye topologie, sal ons eksplisiet die uiteensetting van hierdie benadering tot topologie weergee. Ons de nieer 'n abstrakte konsep wat, in die punt-vrye konteks, ooreenstem met 'n subruimte van 'n topologiese ruimte. Daar sal verder vier voorstellings van hierdie konsep gegee word. Afsluiting, deur middel van operatore, word in die puntvrye konteks ondersoek met behulp van kategorie teorie as taalmedium. Ons sal 'n spesi eke operator in 'n abstrakte konteks de nieer en twee o enskynlik verskillende voorbeelde van hierdie operator verskaf. Daar word dan bewys dat hierdie twee operatore dieselfde is. Kompaktheid is een van die mees belangrikste konsepte in klassieke topologie en as gevolg daarvan geniet dit groot belangstelling onder wiskundiges. 'n Studie in die verwantskap tussen drie swakker forme van kompaktheid word onderneem. Hierdie verwantskap is al in topologie bevestig en goed begryp onder wiskundiges. Dieselfde kan egter, tot 'n mate, nie van die puntvrye konteks ges^e word nie. Ons sal die puntvrye formulering van hierdie swakker konsepte van kompaktheid en hul verbintenis, weergee. 'n Resultaat wat moontlik belowend kan wees, sal ook genoem word.

In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1

1 This thesis concerns separation axioms in point-free topology. We introduce a notion of weak inclusion, which is a relation on a frame that is weaker than the relation ≤. Weak inclusions provide a uniform way to work with standard separation axioms such as subfitness, fitness, and regularity. Proofs using weak inclusions often bring new insight into the nature of the axioms. We focus on results related to the axiom of subfitness. We study a sublocale which is defined as the intersection of all the codense sublocales of a frame. We show that it need not be subfit. For spacial frames, it need not be spacial.