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This book introduces topology and its many applications viewed within a framework that considers metric spaces, continuity, open and closed sets, metric nearness, compactness and completeness. It encompases a complete framework for the study of topology with a variety of applications in science and engineering that include digital image processing, principle component analysis and uniform spaces, stem cell biology, visual merchandising, handwriting forgery detection, paleontology, population dynamics, and medical imaging. 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.