Topology with Applications

Topological Spaces via Near and Far

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.

Contents:

• Basic Framework
• What is Topology?
• Symmetric Proximity
• Continuity and Proximal Continuity
• Separation Axioms
• Uniform Spaces, Filters and Nets
• Compactness and Higher Separation Axioms
• Initial and Final Structures, Embedding
• Grills, Clusters, Bunches and Proximal Wallman Compactification
• Extensions of Continuous Functions: Taimanov Theorem
• Metrisation
• Function Space Topologies
• Hyperspace Topologies
• Selected Topics: Uniformity and Metrisation

Readership: 3rd year undergraduate students, graduate students and researchers in topology; professional and practitioners who are interested in applying topology and its applications especially in science and engineering.
Key Features:

• Complete overview of famous results in topology
• First topology textbook to link proximity space theory (nearness and remoteness) with well-known results in topology
• Presentation of a collection of new applications in a variety of areas such as digital image analysis, stem cell biology, visual merchandising, forgery and paleontology
• The materials in the book have been class-tested over the past thirty years by the authors