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Topology with Applications: Topological Spaces Via Near And TOPOLOGY WITH APPLICATIONS Topological Spaces via Near and Far 8501.9789814407656-tp.indd 1 15/1/13 9:09 AM January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications This page intentionally left blank ii TOPOLOGY WITH APPLICATIONS Topological Spaces via Near and Far Somashekhar A. Naimpally Lakehead University, Canada James F. Peters University of Manitoba, Canada World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8501.9789814407656-tp.indd 2 15/1/13 9:09 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. TOPOLOGY WITH APPLICATIONS Topological Spaces via Near and Far Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4407-65-6 Printed in Singapore. LaiFun - Topology with Applications.pmd 1 1/15/2013, 9:01 AM January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications We dedicate this book to Sudha and Sheela and Saroja v January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications This page intentionally left blank vi January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications Foreword Topological ideas originated in the 19th century, mainly in the works of Riemann and Poincaré, and finally led to the classical definitions of topo- logical space either via the closure operator by Kuratowski, or via open sets. The latter one is the most popular now. However, such progress was preceded by many other attempts and approaches which appear to be useful in different cases and worthwhile to be developed. One such approach is based on the notion of near sets introduced by F. Riesz in 1908. A set X with a nearness relation between its subsets is called a proximity space and every such structure induces a topology on X defined via the closure operator: we say that a point x lies in the closure of a subset U if the subset {x} is near U. It appears that the same topology on X may correspond in this way to different proximities. Moreover many topological results may be inherited from statements concerning proximity spaces. It has to be recalled that in the same manner the proximity structure is induced by the uniform relation introduced by A. Weil. The proximity was rediscovered by V.A. Efremovič in the middle of 1930s and later A.D. Wallace arrived at the same concept. Initially, the proximity theory has developed by its own means with strong contribu- tions by Efremovič and his school and the first author of this book, S.A. Naimpally. Recently, the proximity theory found valuable applications which strongly justify their developments. This book was preceded by a short 1 excellent survey by the same authors of the main ideas of near sets, which 1 Peters, Jim; Naimpally, Som. Applications of near sets. Notices Amer. Math. Soc. 59 (2012), no. 4, 536–542. vii January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications viii Topology with Applications. Topological Spaces via Near and Far have important applications in image processing, image analysis, face recog- nition, and some other engineering and natural science problems. In a sense, the way in which this theory came from its mathematical origin to applications is quite opposite to the development of fuzzy sets arising from applications and leading to a branch of modern set theory. The targets of this book are three-fold. First, it is to expose the main constructions and ideas of set-theoretical topology through the magnifying glass of proximity and uniform spaces. Such an introduction to topology strongly benefits from the simple as well as rigorous notions of near and far at various levels. The motivation originates in advanced calculus and goes to metric spaces, topology, proximity, and uniformity. Diverse results are unified, as for example, in the proximal avatar of the result by A.D. Taimanov on extensions of continuous functions from dense subsets. Second, this book gives an introduction to descriptive proximity rising from extensions of the usual spatial nearness relations in terms of descrip- tively near sets. This leads to many important results that are fundamental building blocks in understanding topology and its implications. Third, this book demonstrates the utility of the notions of near and far in many diverse applications, ranging from cell biology and micropalaeon- tology to camouflage and forgery detection. The applications are made incisive by the accompanying use of near and far in bringing to light the subtleties typically hidden in physical structures such as the distinctively different shapes of microfossils and typical camouflage of animals in natural surroundings. Of particular interest in this book is the range of applications of the proposed approach to topology in study of climate change, mineral and fossil fuel exploration. We strongly recommend this book as a concise and very original in- troduction of set-theoretical topology, nearness theory and their modern applications. October 2012 Professor Iskander A. Taimanov, D. Sc., Member of Russian Academy of Sciences, Chair of Geometry and Topology, Novosibirsk State University, Russia January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications Preface The main purpose in writing this book is to demonstrate the beneficial use of near and far, discovered by F. Riesz over a century ago, from the un- dergraduate to the research level in general topology and its applications. Use of near and far is intuitive yet rigourous at the same time, which is rare in mathematics. The near and far paradigm is based on many years of teaching and research by the authors. The book introduces topology and its many applications viewed within a framework that begins with met- ric spaces and deals with the usual topics such as continuity, open and closed sets, metric nearness, compactness and completeness and glides into topology, proximity and uniformity. Most topics are first studied in metric spaces and later in a topological space. The motivation for this approach is a straightforward, intuitive and yet rigourous rendition of topological concepts. The approach also unifies several scattered results in many ar- eas. Many exercises come from the current literature and some occur in simplified form in metric spaces. The end result is a solid, workable frame- work for the study of topology with a variety of applications in science and engineering that include camouflage, cell biology, digital image anal- ysis, forgery detection, general relativity, microscopy, micropalaeontology, pattern recognition, population dynamics, psychology and visual merchan- dising. We gratefully acknowledge the insightful Foreword by I.A. Taimanov. This Foreword is especially important, since it lucidly brings together the principal highlights of this book and it serves as a commemoration of the seminal work on topology by A.D. Taimanov, who proved one of the most fundamental results in topology concerning extensions of continuous func- tions from dense subspaces. ix January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications This page intentionally left blank x January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications Contents Foreword vii Preface ix 1. Basic Framework 1 1.1 Preliminaries........................... 1 1.2 MetricSpace........................... 2 1.3 GapFunctionalandClosureofaSet............. 3 1.4 LimitofaSequence....................... 6 1.5 Continuity............................. 8 1.6 OpenandClosedSets...................... 10 1.7 MetricandFineProximities.................. 12 1.8 MetricNearness......................... 17 1.9 Compactness........................... 18 1.10 Lindelöf Spaces and Characterisations of Compactness . 21 1.11 Completeness and Total Boundedness ............ 24 1.12 Connectedness.......................... 28 1.13 ChainableMetricSpaces.................... 31 1.14 UCSpaces............................. 32 1.15 FunctionSpaces......................... 33 1.16 Completion............................ 36 1.17 HausdorffMetricTopology................... 38 1.18 First Countable, Second Countable and Separable Spaces 39 1.19 Dense Subspaces and Taimanov’s Theorem . ...... 40 1.20 Application: Proximal Neighbourhoods in Cell Biology . 44 1.21 Problems............................. 44 xi January 15, 2013 9:54 World Scientific Book - 9in x 6in TopologyApplications xii Topology with Applications. Topological Spaces via Near and Far 2. What is Topology? 55 2.1 Topology.............................. 55 2.2 Examples............................. 57 2.3 ClosedandOpenSets...................... 58 2.4 ClosureandInterior....................... 59 2.5 Connectedness.......................... 60 2.6 Subspace .............................. 60 2.7 Bases and Subbases
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