Introduction To Metric And Topological Spaces Sutherland Pdf

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The main reason for taking up such a project is to have an electronic backup of my own handwritten solutions. Mathematics cannot be done without actuallydoing it. However at the undergraduate level many students are put off attempting problems unless they have access to written so-lutions.

Lectures for the course are held at am on Tuesdays in Carslaw and at am on Thursdays in Carslaw Students should also attend a tutorial each week, starting in the 2nd week of the semester. There are two tutorials available, one at 11am on Wednesdays in Carslaw , the other at 2pm E Av 11 am on Wednesdays in Eastern Avenue

Introduction to Metric and Topological Spaces

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Search inside document. Oxford ox2 6D? Oxford University Press is a department of the University of Oxford It furthers the University's objective of excellence in research. Database right Oxford University Press maker First. No part of this publication may be reproduced. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department.

India Printed in Great Britain on acid-free paper by Clays. The address of the companion web site is www. Parts of the first edition have been moved there. This makes room for new material on standard surfaces, intended both to give a brief introduction to geometric topology and also to amplify the section on quotient spaces. Accordingly the numbering in the preface to the first edition no longer applies, although the progression of ideas described there is still roughly followed.

To help convey familiarity, concepts such as closure and interior are introduced first for metric spaces. A colleague who liked other aspects of the first edition complained that his students too readily looked at the answers: now as before I have a concern about students working from this book on their own.

It is a pleasure to thank anonymous referees for their thoughtful sug- gestions for improvements: and equally to thank two distinguished ex-students of New College for the comforting advice to change as lit- tle as possible.

I hope to have steered a middle course in response to all this advice. It is also a pleasure to thank several ex-students and other friends for corrections and improvements to this edition. It is also more than a pleasure to thank Ruth for many things. This book introduces metric and topo- logical spaces by describing some of that influence.

The aim is to move gradually from familiar real analysis to abstract topological spaces; the main topics in the abstract setting are related back to familiar ground as far as possible. Apart from the language of metric and topological spaces, the topics discussed are compactness, connectedness, and completeness.

These form part of the central core of general topology which is now used in several branches of mathematics.

The emphasis is on introduction; the book is not comprehensive even within this central core, and algebraic and geometric topology are not mentioned at all. The stage at which a student of mathematics should see this process of generalization, and the degree of generality he should see, are both controversial.

I have tried to write a book which students can read quite soon after they have had a course on analysis of real-valued functions of one real variable, not necessarily including uniform convergence.

Most readers will find noth- ing new there, but we shall continually refer back to it. With continuity as the motivating concept, the setting is generalized to metric spaces in Chapter 2 and to topological spaces in Chapter 3. The pay-off begins in Chapter 5 with the study of compactness, and continues in later chapters on connectedness and completeness.

In order to introduce uniform con- vergence, Chapter 8 reverts to the traditional approach for real-valued functions of a real variable before interpreting this as convergence in the sup metric. Most of the methods of presentation used are the common property of many mathematicians, but I wish to acknowledge that the way of intro- ducing compactness is influenced by Hewitt It is also a pleasure to acknowledge the influence of many teachers, colleagues, and ex-students on this book, and to thank Peter Strain of the Open University for helpful comments and the staff of the Clarendon Press for their encouragement during the writing.

Oxford, W. Preface vii Preface to reprinted edition Tam grateful to all who have pointed out errors in the first printing even to those who pointed out that the proof of Corollary 1.

In particular, it is a pleasure to thank Roy Dyckhoff, Joan James, and Richard Woolfson for valuable comments and corrections. Ozford, W. Introduction x. More on sets and functions Direct and inverse images Inverse functions 4. Metric spaces Motivation and definition Examples of metric spaces Results about continuous functions on metric spaces Bounded sets in metric spaces Open balls in metric spaces Open sets in metric spaces 6. Continuity in topological spaces; bases Definition Homeomorphisms Bases.

Some concepts in topological spaces Uniform convergence Motivation Definition and examples Cauchy's criterion Uniform limits of sequences Generalizations A set containing just one element is called a singleton set. Intersection and union of sets are denoted by M, U, or , U.

The empty set is written 0. A map or function f we use the terms interchangably between sets X,Y is written f : X Y. We call X the domain of f, and we avoid calling Y anything. In traditional calculus the function f A would not be distinguished from f itself, but when we are being fussy about the precise domains of our functions it is important to make the distinction: f has domain X while f A has domain A.

There are some more concepts relating to sets and functions which we shall focus on in the next chapter. We shall occasionally assume that the terms equivalence relation and countable set are understood. We use N, Z, Q, R, C to denote the sets of positive integers, integers, rational numbers, real numbers, and complex numbers, respectively.

This is our definition of interval—a subset of R is an interval iff it is on the above list. The intervals in i , v , vii and ix are called closed intervals; those in ii , vi , viii and ix are called open in- tervals; and iii , iv are called half-open intervals.

We shall try to avoid the occasional risk of confusing an interval a, 6 in R with a point a, 6 in R? The reader has probably already had practice working with sets; here as revision exercises are a few facts which appear later in the book.

The last two exercises, involving equivalence relations, are relevant to the chap- ter on quotient spaces and only there. They look more complicated than they really are. Exercise 2. This shows that there is a one-one correspondence between equivalence relations on X and partitions of X. In topology the idea of the inverse image of a set under a map is much used, so it is good to be familiar with it. If you are at ease with Definitions 3. If in doubt, skip it now but come back to it later if necessary.

Definition 3. We note immediately that in order to make sense Definition 3. It is the special case of f-! We follow common usage by writing f—! Example 3. Note f-1 1 here is not a singleton set.

The graph of this function is a straight line see Figure 3. Hence f is bijective. Moreover this x is unique for a given y since f is injective. This is a useful alternative, although we shall stick to the narrower interpretation. Of the exercises, 3. More on sets and functions 15 Exercise 3. Exercise 3.

Prove that 90 f -! This gives the next definition some point. Definition 4. Example 4. Examples 4.

Introduction to Metric and Topological Spaces

IICt illJI J-;"' I! Jt i Preface to the first edition One of the ways in which topology has influenced other branches of math- ematics in the past few decades is by putting the study of continuity and convergence into a general setting. This book introduces metric and topo- logical spaces by describing some of that influence. The aim is to move gradually from familiar real analysis to abstract topological spaces; the main topics in the abstract setting are related back to familiar ground as far as possible.


This new edition of Wilson Sutherland's classic text introduces metric and topological spaces by describing some of that influence. The aim is to move gradually.


Introduction to Metric and Topological Spaces by Wilson Sutherland Unofficial Solutions Manual

Topology is the minimal structure on a set of points that allows to define a notion of continuity. We will see how this minimal structure is nevertheless rich enough to build up several other geometric concepts like connectedness or compactness. In contrast, we will also discuss how adding a distance function and thereby turning a topological space into a metric space introduces additional concepts missing in topological spaces, like for example completeness or boundedness.

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Wilson A Sutherland-Introduction to Metric and Topological Spaces-Oxford University Press

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. This is a brief, clearly-written introduction to point-set topology. The approach is axiomatic and abstract — the development is motivated by a desire to generalize properties of the real numbers rather than a need to solve problems from other areas of mathematics. The book assumes some familiarity with the topological properties of the real line, in particular convergence and completeness. The level of abstraction moves up and down through the book, where we start with some real-number property and think of how to generalize it to metric spaces and sometimes further to general topological spaces. Most of the book deals with metric spaces. The book has modest goals.

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. This is a brief, clearly-written introduction to point-set topology. The approach is axiomatic and abstract — the development is motivated by a desire to generalize properties of the real numbers rather than a need to solve problems from other areas of mathematics. The book assumes some familiarity with the topological properties of the real line, in particular convergence and completeness.

Metric and topological spaces

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Wilson A. Second editions of maths textbooks occupy a strange place in the literary universe. Alternatively, they may allow the presentation of the material to be refreshed to reflect the expectations of a new generation of readers.

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An Introduction to Metric and Topological Spaces (Second Edition)

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3 Comments

  1. Nicholas O. 30.05.2021 at 21:08

    Sutherland earned a doctorate at the University of Oxford in under the joint supervision of J.

  2. Curtis B. 31.05.2021 at 10:27

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