On the gluing of quasi-pseudometric spaces

Abstract

A classical problem that arises in geometry is how to glue a family of metric spaces such that the resulting space preserves their properties. In this MSc dissertation, we generalise the concept of gluing a family of metric spaces to the framework of quasi-pseudometric spaces. In particular, we will look at gluing a family of q-hyperconvex quasi-pseudo metric spaces along externally q-hyperconvex subsets and along weakly externally q-hyperconvex subsets such that the resulting space preserves the q-hyperconvexity structure. We relate these results to the well-known results in the literature. We revisit the notion of externally q-hyperconvex quasi-pseudo metric spaces and weakly externally q-hyperconvex spaces and some original results are presented. Moreover, we introduce the concept of gated subsets of a quasi-pseudometric space and extend the notion of strong convexity in our context.