Originally conceived as a text for a one-semester course, it is directed to undergraduate students whose studies of calculus sequence have included definitions and proofs of theorems. The author begins with an informal discussion of set theory in Chapter 1, reserving coverage of countability for Chapter 5, where it appears in the context of compactness. In the second chapter Professor Mendelson discusses metric spaces, paying particular attention to various distance functions which may be defined on Euclidean "n"-space and which lead to the ordinary topology. Chapter 3 takes up the concept of topological space, presenting it as a generalization of the concept of a metric space. Chapters 4 and 5 are devoted to a discussion of the two most important topological properties: connectedness and compactness.
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No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the lishers. Library of Congress Catalog card number: Printed in the United States of America Preface This text is based on lecture notes prepared for a one semester undergraduate course given at Smith College.
The aim has been to present a simple, thorough survey of elementary topics to students whose preparation includes a two-year course in calculus in which some attention has been paid to definitions and proofs of theorems The first chapter consists of the usual discussion of set theory. Since topological space is a generalization of metric space, it is hoped that the reader will observe the similarity, or perhaps redund ancy, in the presentation of these two topics Chapters iv and v are devoted to a discussion of the two most important topological properties, connectedness and com pactness.
The author has borrowed freely from an excellent paper by A. Tucker in the" Proceedings of the First Cana- dian Mathematical Congress"in the section on applications of connectedness. In particular I should like to mention Professors C. Chevalley, S. Eilenberg, I. James, H. Ribeiro, P. Smith, and E. We may, however, illustrate this point by two important examples The set of positive integers or matural numbers is a collec tion of objects N on which there is defined a function 8, called the successor function, satisfying the conditions 1.
There is one and only one object in N, denoted by.
Introduction to Topology
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