LEADER 05043oam 2200529 450 001 9910779692303321 005 20190911112728.0 010 $a1-299-46235-9 010 $a981-4407-66-6 035 $a(OCoLC)840506973 035 $a(MiFhGG)GVRL8RJD 035 $a(EXLCZ)992550000001019234 100 $a20130813h20132013 uy 0 101 0 $aeng 135 $aurun|---uuuua 181 $ctxt 182 $cc 183 $acr 200 10$aTopology with applications $etopological spaces via near and far /$fSomashekhar A. Naimpally, Lakehead University, Canada, James F. Peters, University of Manitoba, Canada 210 $aSingapore $cWorld Scientific Pub. Co.$d2013 210 1$aNew Jersey :$cWorld Scientific,$d[2013] 210 4$d?2013 215 $a1 online resource (xv, 277 pages) $cillustrations (some color) 225 0 $aGale eBooks 300 $aDescription based upon print version of record. 311 $a981-4407-65-8 320 $aIncludes bibliographical references and index. 327 $aForeword; 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 Lindelo?f 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 327 $a1.19 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 Lindelo?f; 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 327 $a3.2 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 327 $a5.4.2 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 327 $a7.6.1 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 327 $a9.9 Problems 330 $aThe 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 606 $aTopology 606 $aProximity spaces 615 0$aTopology. 615 0$aProximity spaces. 676 $a514 700 $aNaimpally$b Somashekhar A$013069 702 $aPeters$b James F. 801 0$bMiFhGG 801 1$bMiFhGG 906 $aBOOK 912 $a9910779692303321 996 $aTopology with applications$93870811 997 $aUNINA