Synopses & Reviews
In topology the three basic concepts of metrics, topologies and uniformities have been treated so far as separate entities by means of different methods and terminology. This is the first book to treat all three as a special case of the concept of approach spaces. This theory provides an answer to natural questions in the interplay between topological and metric spaces by introducing a uniquely well suited supercategory of TOP and MET. The theory makes it possible to equip initial structures of metricizable topological spaces with a canonical structure, preserving the numerical information of the metrics. It provides a solid basis for approximation theory, turning ad hoc notions into canonical concepts, and it unifies topological and metric notions. The book explains the richness of approach structures in great detail; it provides a comprehensive explanation of the categorical set-up, develops the basic theory and provides many examples, displaying links with various areas of mathematics such as approximation theory, probability theory, analysis and hyperspace theory.
"This book is a landmark in the history of general topology....The book is very carefully written, and, with hindsight, the author has presented the material in a simpler manner than that used in his original papers....The book deserves a place on the shelves of mathematicians, especially topologists, who are interested in fundamentals."--Mathematical Review
In topology the three basic concepts of metrics, topologies and uniformities have been treated so far as separate entities by means of different methods and terminology. This work treats all three concepts as a special case of the concept of approach spaces.
Includes bibliographical references (p. -248) and index.
Table of Contents
1. Approach Spaces
2. Topological Approach Spaces
3. Metric Approach Spaces
4. Uniform Approach Spaces
5. Canonical Examples
6. Approach Properties