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On Doitchinov's quietness for arbitrary quasi-uniform spaces
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 Master...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2014
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| _version_ | 1867613303002365952 |
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| access_status_str | Open Access |
| author | Kivuvu, Charly Makitu |
| author2 | Künzi, Hans-Peter A |
| author_browse | Kivuvu, Charly Makitu Künzi, Hans-Peter A |
| author_facet | Künzi, Hans-Peter A Kivuvu, Charly Makitu |
| author_sort | Kivuvu, Charly Makitu |
| collection | Thesis |
| description | 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. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/6665 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:33:59.204Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2014 |
| publishDateRange | 2014 |
| publishDateSort | 2014 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/6665 On Doitchinov's quietness for arbitrary quasi-uniform spaces Kivuvu, Charly Makitu Künzi, Hans-Peter A Mathematics 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. 2014-08-22T10:33:43Z 2014-08-22T10:33:43Z 2010 Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/6665 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics Kivuvu, Charly Makitu On Doitchinov's quietness for arbitrary quasi-uniform spaces |
| thesis_degree_str | Doctoral |
| title | On Doitchinov's quietness for arbitrary quasi-uniform spaces |
| title_full | On Doitchinov's quietness for arbitrary quasi-uniform spaces |
| title_fullStr | On Doitchinov's quietness for arbitrary quasi-uniform spaces |
| title_full_unstemmed | On Doitchinov's quietness for arbitrary quasi-uniform spaces |
| title_short | On Doitchinov's quietness for arbitrary quasi-uniform spaces |
| title_sort | on doitchinov s quietness for arbitrary quasi uniform spaces |
| topic | Mathematics |
| url | http://hdl.handle.net/11427/6665 |
| work_keys_str_mv | AT kivuvucharlymakitu ondoitchinovsquietnessforarbitraryquasiuniformspaces |