The principal aim of this book is to introduce topology and its many applications viewed within a framework that includes a consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.
This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising.
It is the first complete presentation on topology with applications considered in the context of proximity spaces, and the nearness and remoteness of sets of objects. A novel feature throughout this book is the use of near and far, discovered by F Riesz over 100 years ago. In addition, it is the first time that this form of topology is presented in the context of a number of new applications.

Foreword vii

Preface ix

1. Basic Framework 1

1.1 Preliminaries 1

1.2 Metric Space 2

1.3 Gap Functional and Closure of a Set 3

1.4 Limit of a Sequence 6

1.5 Continuity 8

1.6 Open and Closed Sets 10

1.7 Metric and Fine Proximities 12

1.8 Metric Nearness 17

1.9 Compactness 18

1.10 Lindelöf Spaces and Characterisations of Compactness 21

1.11 Completeness and Total Boundedness 24

1.12 Connectedness 28

1.13 Chainable Metric Spaces 31

1.14 UC Spaces 32

1.15 Function Spaces 33

1.16 Completion 36

1.17 Hausdorff Metric Topology 38

1.18 First Countable, Second Countable and Separable Spaces 39

1.19 Dense Subspaces and Taimanov’s Theorem . 40

1.20 Application: Proximal Neighbourhoods in Cell Biology 44

1.21 Problems 44

2. What is Topology? 55

2.1 Topology 55

2.2 Examples 57

2.3 Closed and Open Sets 58

2.4 Closure and Interior 59

2.5 Connectedness 60

2.6 Subspace 60

2.7 Bases and Subbases 61

2.8 More Examples 62

2.9 First Countable, Second Countable and Lindelöf 63

2.10 Application: Topology of Digital Images 64

      2.10.1 Topological Structures in Digital Images 65

      2.10.2 Visual Sets and Metric Topology 65

      2.10.3 Descriptively Remote Sets and Descriptively Near Sets 67

2.11 Problems 69

3. Symmetric Proximity 71

3.1 Proximities 71

3.2 Proximal Neighbourhood 75

3.3 Application: EF-Proximity in Visual Merchandising 75

3.4 Problems 78

4. Continuity and Proximal Continuity 81

4.1 Continuous Functions 81

4.2 Continuous Invariants 83

4.3 Application: Descriptive EF-Proximity in NLO Microscopy 84

      4.3.1 Descriptive L-Proximity and EF-Proximity 85

      4.3.2 Descriptive EF Proximity in Microscope Images 88

4.4 Problems 89

5. Separation Axioms 93

5.1 Discovery of the Separation Axioms 93

5.2 Functional Separation 96

5.3 Observations about EF-Proximity 97

5.4 Application: Distinct Points in Hausdorff Raster Spaces . 97

      5.4.1 Descriptive Proximity 98

      5.4.2 Descriptive Hausdorff Space 102

      5.5 Problems 104

6. Uniform Spaces, Filters and Nets 105

6.1 Uniformity via Pseudometrics 105

6.2 Filters and Ultrafilters 108

6.3 Ultrafilters 111

6.4 Nets (Moore-Smith Convergence) 112

6.5 Equivalence of Nets and Filters 114

6.6 Application: Proximal Neighbourhoods in Camouflage Neighbourhood Filters 114

6.7 Problems 117

7. Compactness and Higher Separation Axioms 119

7.1 Compactness: Net and Filter Views 119

7.2 Compact Subsets . 121

7.3 Compactness of a Hausdorff Space 122

7.4 Local Compactness 123

7.5 Generalisations of Compactness 124

7.6 Application: Compact Spaces in Forgery Detection 125

      7.6.1 Basic Approach in Detecting Forged Handwriting 126

      7.6.2 Roundness and Gradient Direction in Defining Descriptive Point Clusters 128

      7.7 Problems 129

8. Initial and Final Structures, Embedding 131

8.1 Initial Structures 131

8.2 Embedding 133

8.3 Final Structures 134

8.4 Application: Quotient Topology in Image Analysis 134

8.5 Problems 136

9. Grills, Clusters, Bunches and Proximal Wallman Compactification 139

9.1 Grills, Clusters and Bunches 139

9.2 Grills 139

9.3 Clans 140

9.4 Bunches 141

9.5 Clusters 142

9.6 Proximal Wallman Compactification 143

9.7 Examples of Compactifications 145

9.8 Application: Grills in Pattern Recognition . 148

9.9 Problems 156

10. Extensions of Continuous Functions: Taimanov Theorem 157

10.1 Proximal Continuity 157

10.2 Generalised Taimanov Theorem 158

10.3 Comparison of Compactifications 161

10.4 Application: Topological Psychology 161

10.5 Problems 166

11. Metrisation 167

11.1 Structures Induced by a Metric 167

11.2 Uniform Metrisation 168

11.3 Proximal Metrisation 169

11.4 Topological Metrisation 170

11.5 Application: Admissible Covers in Micropalaeontology 172

11.6 Problems 179

12. Function Space Topologies 181

12.1 Topologies and Convergences on a Set of Functions 181

12.2 Pointwise Convergence 182

12.3 Compact Open Topology 184

12.4 Proximal Convergence 187

12.5 Uniform Convergence 188

12.6 Pointwise Convergence and Preservation of Continuity 189

12.7 Uniform Convergence on Compacta 191

12.8 Graph Topologies 193

12.9 Inverse Uniform Convergence for Partial Functions 195

12.10 Application: Hit and Miss Topologies in Population Dynamics 199

12.11 Problems 205

13. Hyperspace Topologies 209

13.1 Overview of Hyperspace Topologies 209

13.2 Vietoris Topology 210

13.3 Proximal Topology 211

13.4 Hausdorff Metric (Uniform) Topology 212

13.5 Application: Local Near Sets in Hawking Chronologies 213

13.6 Problems 217

14. Selected Topics: Uniformity and Metrisation 219

14.1 Entourage Uniformity 219

14.2 Covering Uniformity 222

14.3 Topological Metrisation Theorems 223

14.4 Tietze’s Extension Theorem 227

14.5 Application: Local Patterns 227

      14.5.1 Near Set Approach to Pattern Recognition 228

      14.5.2 Local Metric Patterns 230

      14.5.3 Local Topological Patterns 233

      14.5.4 Local Chronology Patterns 234

      14.5.5 Local Star Patterns 235

      14.5.6 Local Star Refinement Patterns 236

      14.5.7 Local Proximity Patterns 237

14.6 Problems 240

Notes and Further Readings 243

Bibliography 255

Author Index 267

Subject Index 269

中科院文献情报中心