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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/4108 |
| Date | 03 1900 |
| Creators | Masuret, Jacques |
| Contributors | Holgate, D., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 66 p. |
| Rights | University of Stellenbosch |
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