Introduction to Topology
Topology is a large subject with several branches, broadly categorized as algebraic topology, point-set topology, and geometric topology. Point-set topology is the main language for a broad range of mathematical disciplines, while algebraic topology offer
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Introduction to Topology
Introduction to Topology
Tej Bahadur Singh
Introduction to Topology
123
Tej Bahadur Singh Department of Mathematics University of Delhi Delhi, India
ISBN 978-981-13-6953-7 ISBN 978-981-13-6954-4 https://doi.org/10.1007/978-981-13-6954-4
(eBook)
Library of Congress Control Number: 2019933197 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
To the memory of my parents
Preface
This book is intended as an introductory text for senior undergraduate and beginning graduate students of mathematics. Historically, topology has roots scattered in the nineteenth-century works on analysis and geometry. The term “topology” (actually “topologie” in German) was coined by J. B. Listing in 1836. The seminal work of Henri Poincaré (“Analysis Situs” and its five Supplements published during 1895–1909) marks the beginning of the subject of (combinatorial) topology. By the late twenties, topology has evolved as a separate discipline, and it is now a broad subject with many branches, broadly categorized as algebraic topology, general topology (or point-set topology), and geometric topology (or the theory of manifolds). Today, point-set topology is the main language for a wide variety of mathematical disciplines, while algebraic topology serves as a powerful tool for studying the problems in geometry and many other areas of mathematics. The purpose of the present work is to introduce the rudiments of general topology and algebraic topology as a part of the basic vocabulary of mathematics for higher studies. There are some excellent textbooks and treatises on the subject, but they mostly require a level of maturit
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