Uniform spaces

J. R. Isbell0821815121, 9780821815120

Uniform spaces play the same role for uniform continuity as topological spaces for continuity. The theory was created in 1936 by A. Weil, whose original axiomatization was soon followed by those of Bourbaki and Tukey; in this book use is made chiefly of Tukey’s system, based on uniform coverings. The organization of the book as a whole depends on the Eilenberg-MacLane notions of category, functor and naturality, in the spirit of Klein’s Erlanger Program but with greater reach. The preface gives a concise history of the subject since 1936 and a foreword outlines the category theory of Eilenberg and MacLane. The chapters cover fundamental concepts and constructions; function spaces; mappings into polyhedra; dimension (1) and (2); compactifications and locally fine spaces. Most of the chapters are followed by exercises, occasional unsolved problems, and a major unsolved problem; the famous outstanding problem of characterizing the Euclidean plane is discussed in an appendix. There is a good index and a copious bibliography intended not to itemize sources but to guide further reading.

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Table of contents :
Title page ……Page 1
Date-line ……Page 2
PREFACE ……Page 3
TABLE OF CONTENTS ……Page 7
Foreword. Categories ……Page 9
Metric uniform spaces ……Page 13
Uniformities and preuniformities ……Page 15
Uniform topology and uniform continuity ……Page 18
Exercises ……Page 23
Notes ……Page 24
Sum, product, subspace, quotient ……Page 25
Completeness and completion ……Page 29
Compactness and compactification ……Page 32
Proximity ……Page 36
Hyperspace ……Page 39
Exercises ……Page 44
Notes ……Page 47
The functor $U$ ……Page 48
Injective spaces ……Page 51
Equiuniform continuity and semi-uniform products ……Page 55
Closure properties ……Page 61
Exercises ……Page 64
Research Problem B$_1$ ……Page 66
Notes ……Page 67
Uniform complexes ……Page 68
Canonical mappings ……Page 73
Extensions and modifications ……Page 77
Inverse limits ……Page 82
Exercises ……Page 85
Notes ……Page 88
Covering dimension ……Page 90
Extension of mappings ……Page 93
Separation ……Page 97
Metric spaces ……Page 100
Exercises ……Page 104
Research Problem C ……Page 107
Notes ……Page 108
Dimension-preserving compactifications ……Page 109
Examples ……Page 114
Metric case ……Page 118
Freudenthal compactification ……Page 121
Exercises ……Page 129
Notes ……Page 133
The functor $lambda$ ……Page 135
Shirota’s theorem ……Page 139
Products of separable spaces ……Page 142
Glicksberg’s theorem ……Page 145
Supercomplete spaces ……Page 152
Exercises ……Page 153
Notes ……Page 156
Essential coverings ……Page 158
Sum and subset ……Page 160
Coincidence theorems ……Page 165
Exercises ……Page 167
Notes ……Page 169
Appendix. Line and plane ……Page 171
Bibliography ……Page 175
Index ……Page 185