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: 163
1.1 Definition and Examples ......Page 12
1.2 Open Balls and Open Sets ......Page 26
2.1 Convergent Sequences ......Page 46
2.2 Limit and Cluster Points ......Page 50
2.3 Cauchy Sequences and Completeness ......Page 54
2.4 Bounded Sets ......Page 59
2.5 Dense Sets ......Page 61
2.6 Basis ......Page 63
2.7 Boundary of a Set ......Page 64
3.1 Continuous Functions ......Page 67
3.2 Equivalent Definitions of Continuity ......Page 70
3.3 Topological Property ......Page 83
3.4 Uniform Continuity ......Page 86
3.5 Limit of a Function ......Page 90
3.6 Open and closed maps ......Page 91
4.1 Compact Spaces and their Properties ......Page 92
4.2 Continuous Functions on Compact Spaces ......Page 102
4.3 Characterization of Compact Metric Spaces ......Page 106
4.4 Arzela- Ascoli Theorem ......Page 112
5.1 Connected Spaces ......Page 117
5.2 Path Connected spaces ......Page 126
6.1 Examples of Complete Metric Spaces ......Page 133
6.2 Completion of a Metric Space ......Page 142
6.3 Baire Category Theorem ......Page 148
6.4 Banach's Contraction Principle ......Page 154
Bibliography ......Page 160
Index ......Page 161
