Studies in Generalised Metric Spaces

A generalised metric space (∝-metric space) is a metric space where the metric takes its values in an ordered abelian group. In this thesis we give a survey of these spaces, giving particular emphasis to the way in which ordinary metric space theorems have to modified. As this study is a survey, a number of the results are not original but the approach is new. Some results are modifications and/or simplifications of known results. When this is so the original proof will usually be summarised and comparisons made. The main new contributions are the use of the ∝-product to characterize certain ordered abelian groups; the use of the ∝-product in the proof of the topological ∝metrization theorem; and the treatment of the completion of ∝metric spaces. The uniform ∝metrization theorem was proved but it has been discovered that this result has been published recently.
In the first chapter we give an introduction and define the basic concepts and then, in the next two chapters, we develop the theory of convergence and the required topological properties. The fourth chapter is devoted to ordered abelian groups and the last two chapters discuss the important question of metrization (uniform and topological respectively).