Contents

Preface                                                                       vii

Chapter I  Introduction
1.  Euler's theorem                                                             1
2.  Topological equivalence                                                     4
3.  Surfaces                                                                    8
4.  Abstract spaces                                                            12
5.  A classification theorem                                                   16
6.  Topological invariants                                                     19

Chapter 2  Continuity
1.  Open and closed sets                                                       27
2.  Continuous functions                                                       32
3.  A space-filling curve                                                      36
4.  The Tietze extension theorem                                               38

Chapter 3  Compactness and connectedness
1.  Closed bounded subsets of E'                                               43
2.  The Heine-Borel theorem                                                    44
3.  Properties of compact spaces                                               47
4.  Product spaces                                                             51
5.  Connectedness                                                              56
6.  Joining points by paths                                                    61

Chapter 4  Identification spaces
I . Constructing a Moebius strip                                               65
2.  The identification topology                                                66
3.  Topological groups                                                         73
4.  Orbit spaces                                                               78

Chapter 5  The fundamental group
1.  Homotopic maps                                                             87
2.  Construction of the fundamental group                                      92
3.  Calculations                                                               96
4.  Homotopy type                                                             103
5.  The Brouwer fixed-point theorem                                           110
6.  Separation of the plane                                                   112
7.  The boundary of a surface                                                 115

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