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 Thesis / Dissertation en_ZA
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 en_ZA
dc.type.qualificationname PhD en_ZA
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