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.

Reference:
Agyingi, C. 2014. Hyperconvex hulls in catergories of quasi-metric spaces. University of Cape Town.

Includes bibliographical references.