Topology_Without_Tears-S_Morris.pdf

TOPOLOGY WITHOUT TEARS 1

SIDNEY A. MORRIS

Version of June 22, 2001 2

prior written permission from the author.

2 This book is being progressively updated and expanded; it is anticipated that there will

be about ﬁfteen chapters in all. Only those chapters which appear in colour have been

updated so far. If you discover any errors or you have suggested improvements please e-mail:

Sid.Morris@unisa.edu.au

1 c

Contents

Introduction

iii

1 Topological Spaces 1

1.1 Topology ................................. 2

1.2 OpenSets................................. 9

1.3 Finite-ClosedTopology......................... 13

1.4 Postscript................................. 20

Appendix 1: Inﬁnite Sets

Introduction

Topology is an important andinteresting area of mathematics, the study of

which will not only introduce you to new concepts and theorems but also put

into context oldones like continuous functions. However, to say just this is to

understate the signiﬁcance of topology. It is so fundamental that its inﬂuence

is evident in almost every other branch of mathematics. This makes the study

of topology relevant to all who aspire to be mathematicians whether their

ﬁrst love is (or will be) algebra, analysis, category theory, chaos, continuum

mechanics, dynamics, geometry, industrial mathematics, mathematical biology,

mathematical economics, mathematical ﬁnance, mathematical modelling,

mathematical physics, mathematics of communication, number theory,

numerical mathematics, operations research or statistics. Topological notions

like compactness, connectedness and denseness are as basic to mathematicians

of today as sets and functions were to those of last century.

Topology has several diﬀerent branches — general topology (also known

as point-set topology), algebraic topology, diﬀerential topology and topological

algebra — the ﬁrst, general topology, being the door to the study of the others.

We aim in this book to provide a thorough grounding in general topology.

Anyone who conscientiously studies about the ﬁrst ten chapters and solves at

least half of the exercises will certainly have such a grounding.

For the reader who has not previously studied an axiomatic branch of

mathematics such as abstract algebra, learning to write proofs will be a hurdle.

To assist you to learn how to write proofs, quite often in the early chapters, we

include an aside which does not form part of the proof but outlines the thought

process which led to the proof. Asides are indicated in the following manner:

In order to arrive at the proof, we went through this thought process,

which might well be calledthe “discovery” or “experiment phase”.

However, the reader will learn that while discovery or experimentation

is often essential, nothing can replace a formal proof.

iii

INTRODUCTION

There are many exercises in this book. Only by working through a good

number of exercises will you master this course. Very often we include new

concepts in the exercises; the concepts which we consider most important will

generally be introduced again in the text.

Harder exercises are indicated by an *.

Acknowledgment. Portions of earlier versions of this book were usedat

LaTrobe University, University of New England, University of Wollongong,

University of Queensland, University of South Australia and City College of

New York over the last 25 years. I wish to thank those students who criticized

the earlier versions andidentiﬁederrors. Special thanks go to Deborah King for

pointing out numerous errors andweaknesses in the presentation. Thanks also

go to several other colleagues including Carolyn McPhail, Ralph Kopperman,

Rodney Nillsen, Peter Pleasants, Geoﬀrey Prince and Bevan Thompson who

readearlier versions andoﬀeredsuggestions for improvements. Thanks also

go to Jack Gray whose excellent University of New South Wales Lecture Notes

“Set Theory andTransﬁnite Arithmetic”, written in the 1970s, inﬂuencedour

Appendix on Inﬁnite Set Theory.

process without prior written permission from the author.

Chapter 1

Topological Spaces

Introduction

Tennis, football, baseball andhockey may all be exciting games but to play

them you must ﬁrst learn (some of) the rules of the game. Mathematics is no

diﬀerent. So we begin with the rules for topology.

This chapter opens with the deﬁnition of a topology and is then devoted

to some simple examples: ﬁnite topological spaces, discrete spaces, indiscrete

spaces, andspaces with the ﬁnite-closedtopology.

Topology, like other branches of pure mathematics such as group theory, is

an axiomatic subject. We start with a set of axioms andwe use these axioms

to prove propositions andtheorems. It is extremely important to develop your

skill at writing proofs.

Why are proofs so important? Suppose our task were to construct a

building. We would start with the foundations. In our case these are the

axioms or deﬁnitions – everything else is built upon them. Each theorem or

proposition represents a new level of knowledge andmust be ﬁrmly anchoredto

the previous level. We attach the new level to the previous one using a proof.

So the theorems andpropositions are the new heights of knowledge we achieve,

while the proofs are essential as they are the mortar which attaches them to

the level below. Without proofs the structure wouldcollapse.

So what is a mathematical proof? A mathematical proof is a watertight

argument which begins with information you are given, proceeds by logical

argument, andends with what you are askedto prove.

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: