Quasi-metric geometry

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Every time one sees |x-y|, one is looking at a specific metric acting on x and y, whatever they may happen to be, usually numbers or vectors. The notion of the distance between two objects is one of the most fundamental and ubiquitous in many branches of mathematics. A quasi-metric is a generalization of the familiar notion of metric. This dissertation examines what happens in this new setting of quasi-metrics. In particular, in the first chapter we introduce quasi-metrics, provide examples of them, then, given an arbitrary quasi-metric, develop a procedure which allows us to construct a better quasi-metric. Then we look at topological matters, such as openness and continuity. After that, we look at functions on abstract objects called groupoids, which is yet another step toward generality, since the objects we consider here contain the class of quasi-metrics. Dealing with groupoids is useful because it provides a natural structure into which quasi-metrics and quasi-norms fit. After these preliminary chapters, we then introduce linear structure, meaning the quasi-metrics studied are defined on sets in which one can add two points together, and multiply points by numbers, as this is not possible in an abstract set. Next we quantify smoothness of quasi-metric spaces, and throw in measures. For the first six chapters, we worked within a given quasi-metric space, assigning to points the distance between them. The seventh and final chapter deals with the "distance'' between two distinct quasi-metrics.