Gives a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas to encourage geometric thinking and to treat this as a preparatory ground for a general topology course.
Author(s): S. Kumaresan
Publisher: Alpha Science International, Ltd
Year: 2005
Language: English
Pages: 166
TOPOLOGY OF METRIC SPACES......Page 1
Title Page......Page 3
Copyright Page......Page 4
Dedication......Page 6
Preface......Page 8
Contents......Page 12
1.1 Definition and Examples......Page 14
1.2 Open Balls and Open Sets......Page 28
2.1 Convergent Sequences......Page 48
2.2 Limit and Cluster Points......Page 52
2.3 Cauchy Sequences and Completeness......Page 56
2.4 Bounded Sets......Page 61
2.5 Dense Sets......Page 63
2.6 Basis......Page 65
2.7 Boundary of a Set......Page 66
3.1 Continuous Functions......Page 69
3.2 Equivalent Definitions of Continuity......Page 72
3.3 Topological Property......Page 85
3.4 Uniform Continuity......Page 88
3.5 Limit of a Function......Page 92
3.6 Open and closed maps......Page 93
4.1 Compact Spaces and their Properties......Page 94
4.2 Continuous Functions on Compact Spaces......Page 104
4.3 Characterization of Compact Metric Spaces......Page 108
4.4 Arzela–Ascoli Theorem......Page 114
5.1 Connected Spaces......Page 119
5.2 Path Connected spaces......Page 128
6.1 Examples of Complete Metric Spaces......Page 135
6.2 Completion of a Metric Space......Page 144
6.3 Baire Category Theorem......Page 150
6.4 Banach's Contraction Principle......Page 156
Bibliography......Page 162
Index......Page 163
Back Cover......Page 166
