Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad

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Overview

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.

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Product Details

  • ISBN-13: 9780198500308
  • Publisher: Oxford University Press, USA
  • Publication date: 8/28/1997
  • Series: Oxford Mathematical Monographs Series
  • Pages: 262
  • Product dimensions: 6.30 (w) x 9.20 (h) x 0.80 (d)

Table of Contents

Introduction 1
1 Approach Spaces 9
2 Topological Approach Spaces 72
3 Metric Approach Spaces 92
4 Uniform Approach Spaces 117
5 Canonical Examples 136
6 Approach Properties 167
7 Completion 190
8 Compactification 203
App. A.1 Basic Categorical Concepts 218
App. A.2 Topological Constructs 224
App. A.3 Reflective and Coreflective Subconstructs 228
App. B.1 Metric Spaces 231
App. B.2 Probabilistic Metric Spaces 234
App. B.3 Convergence Spaces 236
App. B.4 Quasi-Uniform and Uniform Spaces 238
App. B.5 Proximity Spaces 240
Bibliography 241
Index 249
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