Date of Award
Program or Major
Doctor of Philosophy
As non-Hausdorff spaces are becoming more important in topology, there is a need to consider new notions in topology to supplement the usual structures. This work uses distance functions to find useful generalizations (in a non-Hausdorff context) of the classes of spaces that are important in the Hausdorff setting. We begin, in the first part, with a historical overview that traces the evolution of the notion of distance and its role in the development of general topology.
In the second part of this work, we launch our study of distance functions. Using non-symmetric distance functions, called asymmetric, we generalize the class of symmetrizable spaces which itself includes Moore spaces and metrizable spaces. We also introduce generalizations of Gamma spaces, Nagata spaces and developable spaces. We conclude this work with a number of results about pseudo-metrizable and metrizable spaces.
An underlying theme of this work is that distance functions can provide intuitively-appealing proofs for known theorems that usually have more complex derivations and are often presented with explicit use of the Hausdorff property.
Neveu, Lawrence Valery, "Contributions to the theory of distance functions and its application in general topology" (1997). Doctoral Dissertations. 1952.