Principles of topology

Cover......Page 1
Principles of Topology......Page 4
ISBN-13: 978-81-315-0465-9......Page 5
PREFACE......Page 6
LIST OF SYMBOLS......Page 9
CONTENTS......Page 11
1.1 THE NATURE OF TOPOLOGY......Page 14
1.2 THE ORIGIN OF TOPOLOGY......Page 20
1.3 PRELIMINARY IDEAS FROM SET THEORY......Page 24
1.4 OPERATIONS ON SETS: UNION, INTERSECTION, AND DIFFERENCE......Page 27
1.5 CARTESIAN PRODUCTS......Page 32
1.6 FUNCTIONS......Page 33
1.7 EQUIVALENCE RELATIONS......Page 38
SUGGESTIONS FOR FURTHER READING......Page 41
2.1 UPPER AND LOWER BOUNDS......Page 42
2.2 FINITE AND INFINITE SETS......Page 46
2.3 OPEN SETS AND CLOSED SETS ON THE REAL LINE......Page 52
2.4 THE NESTED INTERVALS THEOREM......Page 59
2.5 THE PLANE......Page 62
SUGGESTIONS FOR FURTHER READING......Page 64
HISTORICAL NOTES I FOR CHAPTER 2......Page 66
3.1 THE DEFINITION AND SOME EXAMPLES......Page 68
3.2 OPEN SETS AND CLOSED SETS IN METRIC SPACES......Page 74
3.3 INTERIOR, CLOSURE, AND BOUNDARY......Page 82
3.4 CONTINUOUS FUNCTIONS......Page 88
3.5 EQUIVALENCE OF METRIC SPACES......Page 91
3.6 NEW SPACES FROM OLD......Page 95
3.7 COMPLETE METRIC SPACES......Page 100
SUGGESTIONS FOR FURTHER READING......Page 109
HISTORICAL NOTES FOR CHAFFER 3......Page 110
4.1 THE DEFINITION AND SOME EXAMPLES......Page 112
4.2 INTERIOR, CLOSURE, AND BOUNDARY......Page 116
4.3 BASIS AND SUBBASIS......Page 122
4.4 CONTINUITY AND TOPOLOGICAL EQUIVALENCE......Page 128
4.5 SUBSPACES......Page 135
SUGGESTIONS FOR FURTHER READING......Page 141
HISTORICAL NOTES FOR CHAPIER 4......Page 142
5.1 CONNECTED AND DISCONNECTED SPACES......Page 144
5.2 THEOREMS ON CONNECTEDNESS......Page 146
5.3 CONNECTED SUBSETS OF THE REAL LINE......Page 156
5.4 APPLICATIONS OF CONNECTEDNESS......Page 157
5.5 PATH CONNECTED SPACES......Page 160
5.6 LOCALLY CONNECTED AND LOCALLY PATH CONNECTED SPACES......Page 167
SUGGESTIONS FOR FURTHER READING......Page 172
HISTORICAL NOTES FOR CHAPTER 5......Page 173
6.1 COMPACT SPACES AND SUBSPACES......Page 174
6.2 COMPACTNESS AND CONTINUITY......Page 180
6.3 PROPERTIES RELATED TO COMPACTNESS......Page 185
6.4 ONE-POINT COMPACTIFICATION......Page 194
6.5 THE CANTOR SET......Page 200
SUGGESTIONS FOR FURTHER READING......Page 205
HISTORICAL NOTES FOR CHAFFER 6......Page 206
7.1 FINITE PRODUCTS......Page 208
7.2 ARBITRARY PRODUCTS......Page 217
7.3 COMPARISON OF TOPOLOGIES......Page 224
7.4 QUOTIENT SPACES......Page 226
7.5 SURFACES AND MANIFOLDS......Page 234
SUGGESTIONS FOR FURTHER READING......Page 241
HISTORICAL NOTES FOR CHAFFER 7......Page 242
8.1 T0, T1, AND T2-SPACES......Page 244
8.2 REGULAR SPACES......Page 247
8.3 NORMAL SPACES......Page 250
8.4 SEPARATION BY CONTINUOUS FUNCTIONS......Page 256
8.5 METRIZATION......Page 266
8.6 THE STONE-CECH COMPACTIFICATION......Page 274
SUGGESTIONS FOR FURTHER READING......Page 278
HISTORICAL NOTES FOR CHAFFER 8......Page 279
9.2 THE FUNDAMENTAL GROUP......Page 280
9.3 THE FUNDAMENTAL GROUP OF S^1......Page 293
9.4 ADDITIONAL EXAMPLES OF FUNDAMENTAL GROUPS......Page 299
9.5 THE BROUWER FIXED POINT THEOREM AND RELATED RESULTS......Page 304
9.6 CATEGORIES AND FUNCTORS......Page 308
SUGGESTIONS FOR FURTHER READING......Page 311
HISTORICAL NOTES FOR CHAFFER 9......Page 312
Appendix: Introduction to Groups......Page 314
BIBLIOGRAPHY......Page 316
INDEX......Page 318