Contents -- Foreword -- Preface -- 1. Basic Framework -- 1.1 Preliminaries -- 1.2 Metric Space -- 1.3 Gap Functional and Closure of a Set -- 1.4 Limit of a Sequence -- 1.5 Continuity -- 1.6 Open and Closed Sets -- 1.7 Metric and Fine Proximities -- 1.8 Metric Nearness -- 1.9 Compactness -- 1.10 Lindelöf Spaces and Characterisations of Compactness -- 1.11 Completeness and Total Boundedness -- 1.12 Connectedness -- 1.13 Chainable Metric Spaces -- 1.14 UC Spaces -- 1.15 Function Spaces -- 1.16 Completion -- 1.17 Hausdorff Metric Topology -- 1.18 First Countable, Second Countable and Separable Spaces -- 1.19 Dense Subspaces and Taimanov's Theorem -- 1.20 Application: Proximal Neighbourhoods in Cell Biology -- 1.21 Problems -- 2. What is Topology? -- 2.1 Topology -- 2.2 Examples -- 2.3 Closed and Open Sets -- 2.4 Closure and Interior -- 2.5 Connectedness -- 2.6 Subspace -- 2.7 Bases and Subbases -- 2.8 More Examples -- 2.9 First Countable, Second Countable and Lindelöf -- 2.10 Application: Topology of Digital Images -- 2.10.1 Topological Structures in Digital Images -- 2.10.2 Visual Sets and Metric Topology -- 2.10.3 Descriptively Remote Sets and Descriptively Near Sets -- 2.11 Problems -- 3. Symmetric Proximity -- 3.1 Proximities -- 3.2 Proximal Neighbourhood -- 3.3 Application: EF-Proximity in Visual Merchandising -- 3.4 Problems -- 4. Continuity and Proximal Continuity -- 4.1 Continuous Functions -- 4.2 Continuous Invariants -- 4.3 Application: Descriptive EF-Proximity in NLO Microscopy -- 4.3.1 Descriptive L-Proximity and EF-Proximity -- 4.3.2 Descriptive EF Proximity in Microscope Images -- 4.4 Problems -- 5. Separation Axioms -- 5.1 Discovery of the Separation Axioms -- 5.2 Functional Separation -- 5.3 Observations about EF-Proximity -- 5.4 Application: Distinct Points in Hausdor. Raster Spaces -- 5.4.1 Descriptive Proximity.

5.4.2 Descriptive Hausdorff Space -- 5.5 Problems -- 6. Uniform Spaces, Filters and Nets -- 6.1 Uniformity via Pseudometrics -- 6.2 Filters and Ultrafilters -- 6.3 Ultrafilters -- 6.4 Nets (Moore-Smith Convergence) -- 6.5 Equivalence of Nets and Filters -- 6.6 Application: Proximal Neighbourhoods in Camouflage Neighbourhood Filters -- 6.7 Problems -- 7. Compactness and Higher Separation Axioms -- 7.1 Compactness: Net and Filter Views -- 7.2 Compact Subsets -- 7.3 Compactness of a Hausdorff Space -- 7.4 Local Compactness -- 7.5 Generalisations of Compactness -- 7.6 Application: Compact Spaces in Forgery Detection -- 7.6.1 Basic Approach in Detecting Forged Handwriting -- 7.6.2 Roundness and Gradient Direction in Defining Descriptive Point Clusters -- 7.7 Problems -- 8. Initial and Final Structures, Embedding -- 8.1 Initial Structures -- 8.2 Embedding -- 8.3 Final Structures -- 8.4 Application: Quotient Topology in Image Analysis -- 8.5 Problems -- 9. Grills, Clusters, Bunches and Proximal Wallman Compactification -- 9.1 Grills, Clusters and Bunches -- 9.2 Grills -- 9.3 Clans -- 9.4 Bunches -- 9.5 Clusters -- 9.6 Proximal Wallman Compactification -- 9.7 Examples of Compactifications -- 9.8 Application: Grills in Pattern Recognition -- 9.9 Problems -- 10. Extensions of Continuous Functions: Taimanov Theorem -- 10.1 Proximal Continuity -- 10.2 Generalised Taimanov Theorem -- 10.3 Comparison of Compactifications -- 10.4 Application: Topological Psychology -- 10.5 Problems -- 11. Metrisation -- 11.1 Structures Induced by aMetric -- 11.2 Uniform Metrisation -- 11.3 Proximal Metrisation -- 11.4 Topological Metrisation -- 11.5 Application: Admissible Covers in Micropalaeontology -- 11.6 Problems -- 12. Function Space Topologies -- 12.1 Topologies and Convergences on a Set of Functions -- 12.2 Pointwise Convergence -- 12.3 Compact Open Topology.

12.4 Proximal Convergence -- 12.5 Uniform Convergence -- 12.6 Pointwise Convergence and Preservation of Continuity -- 12.7 Uniform Convergence on Compacta -- 12.8 Graph Topologies -- 12.9 Inverse Uniform Convergence for Partial Functions -- 12.10 Application: Hit and Miss Topologies in Population Dynamics -- 12.11 Problems -- 13. Hyperspace Topologies -- 13.1 Overview of Hyperspace Topologies -- 13.2 Vietoris Topology -- 13.3 Proximal Topology -- 13.4 Hausdorff Metric (Uniform) Topology -- 13.5 Application: Local Near Sets in Hawking Chronologies -- 13.6 Problems -- 14. Selected Topics: Uniformity and Metrisation -- 14.1 Entourage Uniformity -- 14.2 Covering Uniformity -- 14.3 Topological Metrisation Theorems -- 14.4 Tietze's Extension Theorem -- 14.5 Application: Local Patterns -- 14.5.1 Near Set Approach to Pattern Recognition -- 14.5.2 Local Metric Patterns -- 14.5.3 Local Topological Patterns -- 14.5.4 Local Chronology Patterns -- 14.5.5 Local Star Patterns -- 14.5.6 Local Star Refinement Patterns -- 14.5.7 Local Proximity Patterns -- 14.6 Problems -- Notes and Further Readings -- Bibliography -- Subject Index.