Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad (Oxford Mathematical Monographs)

Description

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. Preface Introduction 1 Approach Spaces 1.1 Distances 1.2 Limit Operators 1.3 Approach Systems 1.4 Gauges 1.5 Towers 1.6 Hull Operators 1.7 Regular Function Frames 1.8 Approach Spaces 1.9 Contractions 1.10 The Topological Construct Ap 2 Topological Approach Spaces 2.1 Topological Approach Structures 2.2 The Embedding Of Top Ln Ap 2.3 Bireflective Bicoreflective Subconstructs 3 Metric Approach Spaces 3.1 Metric Approach Structures 3.2 Convergence Ln Metric Approach Spaces 3.3 The Embedding Of Pqmetoo Ln Ap 3.4 The Epireflective Hull Of Pqmetoo 4 Uniform Approach Spaces 4.1 The Epireflective Hull Of Pmetoo 4.2 Convergence Ln Uap 4.3 Categorical Aspects 4.4 Relationship With Uniform Spaces 5 Canonical Examples 5.1 Spaces Of Measures 5.2 Functlon Spaces 5.3 Hyperspaces 5.4 Probabilistic Metric Spaces 5.5 Spaces Of Random Variables 6 Approach Properties 6.1 Compactness 6.2 Connectedness 6.3 Completeness 7 Completion 7.1 Construction 7.2 Example: Function Spaces 8 Compactification 8.1 Construction 8.2 Example: The Case Of βN Appendix A A.L Basic Categorical Concepts A.2 Topological Constructs A.3 Reflective And Coreflective Subconstructs Appendix B B.L Metric Spaces B.2 Probabilistic Metric Spaces B.3 Convergence Spaces B.4 Quasi-Uniform And Uniform Spaces B.5 Proximity Spaces Bibliography Index