TOPOLOGY WITH APPLICATIONS

Topological Spaces via Near and Far

Somashekhar A. Naimpally Lakehead University, Canada

James F. Peters University of Manitoba, Canada

YJ? World Scientific

• • • NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI • HONG KONG • TAIPEI • CHENNAI Contents

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 Lindelof and Characterisations of . Spaces 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 Proximal in Application: Neighbourhoods Cell Biology . . 44 1.21 Problems 44

xi xii Topology with Applications. Topological Spaces via Near and Far

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 Lindelof 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 Contents xiii

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 Ill

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 in Application: Compact Spaces 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 xiv Topology with Applications. Topological Spaces via Near and Far

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 Contents xv

13.3 Proximal Topology 211 13.4 Hausdorff Metric (Uniform) Topology 212 13.5 Local Near Sets in Application: 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