Hyperconvex hulls in catergories of quasi-metric spaces

 

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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.identifier.citation Agyingi, C. 2014. Hyperconvex hulls in catergories of quasi-metric spaces. University of Cape Town. en_ZA
dc.identifier.uri http://hdl.handle.net/11427/12708
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.language.iso eng en_ZA
dc.title Hyperconvex hulls in catergories of quasi-metric spaces en_ZA
dc.type Doctoral Thesis
uct.type.publication Research en_ZA
uct.type.resource Thesis en_ZA
dc.publisher.institution University of Cape Town
dc.publisher.faculty Faculty of Science en_ZA
dc.publisher.department Department of Mathematics and Applied Mathematics en_ZA
dc.type.qualificationlevel Doctoral
dc.type.qualificationname PhD en_ZA
uct.type.filetype Text
uct.type.filetype Image
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.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.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 T&#8320;-quasi-metric space X has a q-hyperconvex hull QX , which is joincompact if X is joincompact. They called a T&#8320;-quasi-metric space q-hyperconvex if and only if it is injective in the category of T&#8320;-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 T&#8320;-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 T&#8320;-quasi-metric spaces and T&#8320;-ultra-quasimetric spaces. We show in particular that most of the results stay the same as we move from T&#8320;-quasi-metric spaces to T&#8320;-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 &#915;-ultra-quasi-metric space, for an arbitrary countable set &#915; 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


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