Hyperconvex hulls in catergories of quasi-metric spaces
| dc.contributor.advisor | Künzi, Hans-Peter A | en_ZA |
| dc.contributor.author | Agyingi, Collins Amburo | en_ZA |
| dc.date.accessioned | 2015-05-04T07:04:08Z | |
| dc.date.available | 2015-05-04T07:04:08Z | |
| dc.date.issued | 2014 | en_ZA |
| dc.description | Includes bibliographical references. | en_ZA |
| dc.description.abstract | Isbell showed that every metric space has an injective hull, that is, every metric space has a “minimal” hyperconvex metric superspace. Dress then showed that the hyperconvex hull is a tight extension. In analogy to Isbell’s theory Kemajou et al. proved that each T₀-quasi-metric space X has a q-hyperconvex hull QX , which is joincompact if X is joincompact. They called a T₀-quasi-metric space q-hyperconvex if and only if it is injective in the category of T₀-quasi-metric spaces and non-expansive maps. Agyingi et al. generalized results due to Dress on tight extensions of metric spaces to the category of T₀-quasi-metric spaces and non-expansive maps. In this dissertation, we shall study tight extensions (called uq-tight extensions in the following) in the categories of T₀-quasi-metric spaces and T₀-ultra-quasimetric spaces. We show in particular that most of the results stay the same as we move from T₀-quasi-metric spaces to T₀-ultra-quasi-metric spaces. We shall show that these extensions are maximal among the uq-tight extensions of the space in question. In the second part of the dissertation we shall study the q-hyperconvex hull by viewing it as a space of minimal function pairs. We will also consider supseparability of the space of minimal function pairs. Furthermore we study a special subcollection of bicomplete supseparable quasi-metric spaces: bicomplete supseparable ultra-quasi-metric spaces. We will show the existence and uniqueness (up to isometry) of a Urysohn Γ-ultra-quasi-metric space, for an arbitrary countable set Γ of non-negative real numbers including 0. | en_ZA |
| dc.identifier.apacitation | Agyingi, C. A. (2014). <i>Hyperconvex hulls in catergories of quasi-metric spaces</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/12708 | en_ZA |
| dc.identifier.chicagocitation | Agyingi, Collins Amburo. <i>"Hyperconvex hulls in catergories of quasi-metric spaces."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2014. http://hdl.handle.net/11427/12708 | en_ZA |
| dc.identifier.citation | Agyingi, C. 2014. Hyperconvex hulls in catergories of quasi-metric spaces. University of Cape Town. | en_ZA |
| dc.identifier.ris | TY - Thesis / Dissertation AU - Agyingi, Collins Amburo AB - Isbell showed that every metric space has an injective hull, that is, every metric space has a “minimal” hyperconvex metric superspace. Dress then showed that the hyperconvex hull is a tight extension. In analogy to Isbell’s theory Kemajou et al. proved that each T₀-quasi-metric space X has a q-hyperconvex hull QX , which is joincompact if X is joincompact. They called a T₀-quasi-metric space q-hyperconvex if and only if it is injective in the category of T₀-quasi-metric spaces and non-expansive maps. Agyingi et al. generalized results due to Dress on tight extensions of metric spaces to the category of T₀-quasi-metric spaces and non-expansive maps. In this dissertation, we shall study tight extensions (called uq-tight extensions in the following) in the categories of T₀-quasi-metric spaces and T₀-ultra-quasimetric spaces. We show in particular that most of the results stay the same as we move from T₀-quasi-metric spaces to T₀-ultra-quasi-metric spaces. We shall show that these extensions are maximal among the uq-tight extensions of the space in question. In the second part of the dissertation we shall study the q-hyperconvex hull by viewing it as a space of minimal function pairs. We will also consider supseparability of the space of minimal function pairs. Furthermore we study a special subcollection of bicomplete supseparable quasi-metric spaces: bicomplete supseparable ultra-quasi-metric spaces. We will show the existence and uniqueness (up to isometry) of a Urysohn Γ-ultra-quasi-metric space, for an arbitrary countable set Γ of non-negative real numbers including 0. DA - 2014 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2014 T1 - Hyperconvex hulls in catergories of quasi-metric spaces TI - Hyperconvex hulls in catergories of quasi-metric spaces UR - http://hdl.handle.net/11427/12708 ER - | en_ZA |
| dc.identifier.uri | http://hdl.handle.net/11427/12708 | |
| dc.identifier.vancouvercitation | Agyingi CA. Hyperconvex hulls in catergories of quasi-metric spaces. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2014 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/12708 | en_ZA |
| dc.language.iso | eng | en_ZA |
| dc.publisher.department | Department of Mathematics and Applied Mathematics | en_ZA |
| dc.publisher.faculty | Faculty of Science | en_ZA |
| dc.publisher.institution | University of Cape Town | |
| dc.title | Hyperconvex hulls in catergories of quasi-metric spaces | en_ZA |
| dc.type | Doctoral Thesis | |
| dc.type.qualificationlevel | Doctoral | |
| dc.type.qualificationname | PhD | en_ZA |
| uct.type.filetype | Text | |
| uct.type.filetype | Image | |
| uct.type.publication | Research | en_ZA |
| uct.type.resource | Thesis | en_ZA |
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