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Closure and compactness in frames
Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : University of Stellenbosch
2010
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| _version_ | 1867614023587987456 |
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| access_status_str | Open Access |
| author | Masuret, Jacques |
| author2 | Holgate, D. |
| author_browse | Holgate, D. Masuret, Jacques |
| author_facet | Holgate, D. Masuret, Jacques |
| author_sort | Masuret, Jacques |
| collection | Thesis |
| dc_rights_str_mv | University of Stellenbosch |
| description | Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/4108 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:45:26.037Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2010 |
| publishDateRange | 2010 |
| publishDateSort | 2010 |
| publisher | Stellenbosch : University of Stellenbosch |
| publisherStr | Stellenbosch : University of Stellenbosch |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/4108 Closure and compactness in frames Masuret, Jacques Holgate, D. University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Closure Compactness Frames Point-free topology Dissertations -- Mathematics Theses -- Mathematics Topological space 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. 2010-02-24T14:11:48Z 2010-08-13T14:59:09Z 2010-02-24T14:11:48Z 2010-08-13T14:59:09Z 2010-03 Thesis http://hdl.handle.net/10019.1/4108 en University of Stellenbosch 66 p. application/pdf Stellenbosch : University of Stellenbosch |
| spellingShingle | Closure Compactness Frames Point-free topology Dissertations -- Mathematics Theses -- Mathematics Topological space Masuret, Jacques Closure and compactness in frames |
| title | Closure and compactness in frames |
| title_full | Closure and compactness in frames |
| title_fullStr | Closure and compactness in frames |
| title_full_unstemmed | Closure and compactness in frames |
| title_short | Closure and compactness in frames |
| title_sort | closure and compactness in frames |
| topic | Closure Compactness Frames Point-free topology Dissertations -- Mathematics Theses -- Mathematics Topological space |
| url | http://hdl.handle.net/10019.1/4108 |
| work_keys_str_mv | AT masuretjacques closureandcompactnessinframes |