{"id":{"repo_id":"cape-town","oai_identifier":"oai:open.uct.ac.za:11427/9572"},"canonical_url":"https://search.dev.ndltd.org/etd/cape-town/oai:open.uct.ac.za:11427/9572","repository":{"repo_id":"cape-town","name":"University of Cape Town","base_url":"https://open.uct.ac.za/oai/request"},"display":{"title":"E-compactness in pointfree topology","abstract":"The main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames.","abstract_html":"The main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames.","abstract_has_math":false,"creators":["Marcus, Nizar"],"institution":"Department of Mathematics and Applied Mathematics","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Gilmour, Christopher Robert Anderson"],"committee_chairs":[],"committee_members":[],"year":1998,"date_issued":"1998","date_published":"1998","updated_at":"2026-07-22T22:23:19Z","subjects":[],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11427/9572","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gilmour, Christopher Robert Anderson"]},{"key":"dc:creator","label":"Author","values":["Marcus, Nizar"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-11-11T20:11:33Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-11-11T20:11:33Z"]},{"key":"dc:date.issued","label":"Date","values":["1998"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Applied Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cape Town"]},{"key":"dc:type","label":"Dc Type","values":["Doctoral Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11427/9572"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Bibliography: leaves 100-107."]},{"key":"dc:description.abstract","label":"Abstract","values":["The main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames."]},{"key":"dc:title","label":"Title","values":["E-compactness in pointfree topology"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gilmour, Christopher Robert Anderson"],"dc:creator":["Marcus, Nizar"],"dc:date.accessioned":["2014-11-11T20:11:33Z"],"dc:date.available":["2014-11-11T20:11:33Z"],"dc:date.issued":["1998"],"dc:description":["Bibliography: leaves 100-107."],"dc:description.abstract":["The main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames."],"dc:identifier.uri":["http://hdl.handle.net/11427/9572"],"dc:language.iso":["eng"],"dc:publisher.department":["Department of Mathematics and Applied Mathematics"],"dc:publisher.institution":["University of Cape Town"],"dc:title":["E-compactness in pointfree topology"],"dc:type":["Doctoral Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-22T22:23:19Z"}