On Doitchinov's quietness for arbitrary quasi-uniform spaces

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dc.contributor.advisorKünzi, Hans-Peter Aen_ZA
dc.contributor.authorKivuvu, Charly Makituen_ZA
dc.date.accessioned2014-08-22T10:33:43Z
dc.date.available2014-08-22T10:33:43Z
dc.date.issued2010en_ZA
dc.description.abstractThe notion of a quiet quasi-uniform space was introduced by Doitchinov in1988 when he developed an interesting completion theory for this class ofquasi-uniform spaces. At the same time Doitchinov developed a similar com-pletion theory for a class of T0-balanced quasi-pseudometric spaces.In my Masters thesis I showed that the Doitchinov completion theory forbalanced quasi-pseudometric spaces can be extended to arbitrary T0-quasi-pseudometric spaces. That completion was called the B-completion.The principal aim of this thesis is to investigate whether the Doitchinov com-pletion theory for quiet quasi-uniform space can be extended to arbitraryT0-quasi-uniform spaces. The main result in this thesis is negative and leadsus to conclude that Doitchinov's completion theory for quiet quasi-uniformspaces cannot be fully extended to arbitrary quasi-uniform spaces, becauseinvestigations due to De¶ak indicate that no suitable concept of a quiet Cauchy¯lter pair exists which could replace the quasi-pseudometric concept of a bal-anced Cauchy ¯lter pair in the quasi-uniform setting. Under these circum-stances we therefore suggest that in an arbitrary quasi-uniform space, weshould work with a nonempty subbasic family of quasi-pseudometrics and anappropriate concept of balancedness of Cauchy ¯lter pairs with respect tothat family. In this way we obtain a general theory of the B-completion fora subbasic T0-family of quasi-pseudometrics that can be applied to the studyof any quasi-uniform space.en_ZA
dc.identifier.apacitationKivuvu, C. M. (2010). <i>On Doitchinov's quietness for arbitrary quasi-uniform spaces</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/6665en_ZA
dc.identifier.chicagocitationKivuvu, Charly Makitu. <i>"On Doitchinov's quietness for arbitrary quasi-uniform spaces."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2010. http://hdl.handle.net/11427/6665en_ZA
dc.identifier.citationKivuvu, C. 2010. On Doitchinov's quietness for arbitrary quasi-uniform spaces. University of Cape Town.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Kivuvu, Charly Makitu AB - The notion of a quiet quasi-uniform space was introduced by Doitchinov in1988 when he developed an interesting completion theory for this class ofquasi-uniform spaces. At the same time Doitchinov developed a similar com-pletion theory for a class of T0-balanced quasi-pseudometric spaces.In my Masters thesis I showed that the Doitchinov completion theory forbalanced quasi-pseudometric spaces can be extended to arbitrary T0-quasi-pseudometric spaces. That completion was called the B-completion.The principal aim of this thesis is to investigate whether the Doitchinov com-pletion theory for quiet quasi-uniform space can be extended to arbitraryT0-quasi-uniform spaces. The main result in this thesis is negative and leadsus to conclude that Doitchinov's completion theory for quiet quasi-uniformspaces cannot be fully extended to arbitrary quasi-uniform spaces, becauseinvestigations due to De¶ak indicate that no suitable concept of a quiet Cauchy¯lter pair exists which could replace the quasi-pseudometric concept of a bal-anced Cauchy ¯lter pair in the quasi-uniform setting. Under these circum-stances we therefore suggest that in an arbitrary quasi-uniform space, weshould work with a nonempty subbasic family of quasi-pseudometrics and anappropriate concept of balancedness of Cauchy ¯lter pairs with respect tothat family. In this way we obtain a general theory of the B-completion fora subbasic T0-family of quasi-pseudometrics that can be applied to the studyof any quasi-uniform space. DA - 2010 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2010 T1 - On Doitchinov's quietness for arbitrary quasi-uniform spaces TI - On Doitchinov's quietness for arbitrary quasi-uniform spaces UR - http://hdl.handle.net/11427/6665 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/6665
dc.identifier.vancouvercitationKivuvu CM. On Doitchinov's quietness for arbitrary quasi-uniform spaces. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2010 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/6665en_ZA
dc.language.isoengen_ZA
dc.publisher.departmentDepartment of Mathematics and Applied Mathematicsen_ZA
dc.publisher.facultyFaculty of Scienceen_ZA
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherMathematicsen_ZA
dc.titleOn Doitchinov's quietness for arbitrary quasi-uniform spacesen_ZA
dc.typeDoctoral Thesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnamePhDen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
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