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Record Nr. |
UNINA9910814647703321 |
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Autore |
Naimpally Somashekhar A |
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Titolo |
Topology with applications : topological spaces via near and far / / Somashekhar A. Naimpally, Lakehead University, Canada, James F. Peters, University of Manitoba, Canada |
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Pubbl/distr/stampa |
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Singapore, : World Scientific Pub. Co., 2013 |
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New Jersey : , : World Scientific, , [2013] |
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�2013 |
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ISBN |
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1-299-46235-9 |
981-4407-66-6 |
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Descrizione fisica |
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1 online resource (xv, 277 pages) : illustrations (some color) |
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Collana |
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Disciplina |
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Soggetti |
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Topology |
Proximity spaces |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Description based upon print version of record. |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Nota di contenuto |
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Foreword; Preface; Contents; 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 Theorem1.20 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 |
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3.2 Proximal Neighbourhood3.3 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 Space5.5 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 Handwriting7.6.2 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 |
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Sommario/riassunto |
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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, f |
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