This tract provides a compact introduction to the theory of proximity spaces and their generalisations, making the subject accessible to readers having a basic knowledge of topological and uniform spaces, such as can be found in standard textbooks. Two chapters are devoted to fundamentals, the main result being the proof of the existence of the Smirnov compactification using clusters. Chapter 3 discusses the interrelationships between proximity spaces and uniform spaces and contains some of the most interesting results in the theory of proximity spaces. The final chapter introduces the reader to several generalised forms of proximity structures and studies one of them in detail. The bibliography contains over 130 references to the scattered research literature on proximity spaces, in addition to general references.
Author(s): S. A. Naimpally
Series: Cambridge Tracts in Mathematics
Publisher: CUP
Year: 1970
Language: English
Pages: 139
Cover......Page 1
Cambridge Tracts in Mathematics and Mathematical Physics 59......Page 2
Proximity Spaces......Page 4
9780521091831......Page 5
Contents......Page 6
Preface......Page 8
Index of Notations......Page 10
Historical Background......Page 12
1 Introduction......Page 18
2 Topology induced by a proximity......Page 21
3 Alternate description of proximity......Page 26
4 Subspaces and products of proximity spaces......Page 30
Notes......Page 37
5 Clusters and ultrafilters......Page 38
6 Duality in proximity spaces......Page 45
7 Smirnov compactification......Page 49
8 Proximity weight and compactification......Page 58
9 Local proximity spaces......Page 63
Notes......Page 72
10 Proximity induced by a uniformity......Page 74
11 Completion of a uniform space by Cauchy clusters......Page 78
12 Proximity class of uniformities......Page 82
13 Generalized uniform structures......Page 89
14 Proximity and height......Page 95
15 Hyperspace uniformities......Page 98
Notes......Page 103
16 Proximal convergence......Page 105
17 Unified theories of topology, proximity and uniformity......Page 108
18 Sequential proximity......Page 111
19 Generalized proximities......Page 115
20 More on Lodato spaces......Page 119
Notes......Page 125
General References......Page 127
Bibliography for proximity spaces......Page 128
Index......Page 136
