Introduction to metric spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Edinburgh
Oliver & Boyd
1972
|
| Series: | University mathematical texts
42 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 198 S. Ill. |
| ISBN: | 0050024531 |
Staff View
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Record in the Search Index
| _version_ | 1804121399706517504 |
|---|---|
| adam_text | CONTENTS
CHAPTER 1 PRELIMINARIES
1.1 Sets 1
1.2 Functions 3
1.3 Equivalence relations 6
1.4 Countability 7
1.5 The real number system 7
1.6 Concerning continuous functions and Riemann
integration 11
1.7 Negation 16
1.8 Minkowski s inequalities 17
CHAPTER 2 BASIC TERMINOLOGY
2.1 Definition of a metric space 21
2.2 Examples of metric spaces 24
2.3 Convergence 27
2.4 Examples concerning convergence in metric
spaces 29
2.5 The basic topological concepts 32
2.6 Open and closed sets: introduction 36
2.7 Open and closed sets: continued 39
2.8 Relative openness 43
2.9 Cluster values of a sequence 45
2.10 Denseness 47
2.11 The boundary, interior, exterior of a set 48
CHAPTER 3 CONTINUOUS FUNCTIONS
3.1 Introduction 51
3.2 Topological characterizations of continuity 54
3.3 Homeomorphisms 58
3.4 Equivalent metrics 60
3.5 The distances p(x. Y). p(Y.Z) 64
CHAPTER 4 COMPLETENESS
4.1 Complete metric spaces 66
4.2 Examples concerning the completeness of metric
spaces 69
4.3 Cantor s intersection theorem 73
4.4 The completion of incomplete metric spaces 76
4.5 The uniqueness of (X, p) up to isometry 82
4.6 The contraction mapping theorem 84
4.7 Simple applications of the contraction mapping
theorem 88
4.8 Picard s theorem 92
4.9 An implicit function theorem 97
vii
viii CONTENTS
CHAPTER 5 COMPACTNESS
5.1 Introduction 99
5.2 The definitions 102
5.3 The first equivalent characterization of
compactness 104
5.4 The second equivalent characterization of
compactness 108
5.5 The third equivalent characterization of
compactness 109 ,
5.6 Summary of equivalent characterizations of
compactness 110
5.7 Compact subsets 113
5.8 Properties of functions continuous over compact
sets 117
5.9 Examples concerning compactness in metric
spaces 121
5.10 A further example: the Arzela Ascoli theorem 124
5.11 Peano s theorem 129
CHAPTER 6 CONNECTEDNESS
6.1 Introduction and definitions 132
6.2 Connected and disconnected spaces 134 i,
6.3 The connected sets of the real line 138
6.4 The generalization of the intermediate value
theorem 140
6.5 Pathwise connectedness 142
6.6 The components of a disconnected space 146
CHAPTER 7 FURTHER TOPICS
A. Extension theorems
7.1 Introduction ; limits 148
7.2 Two extension theorems 151
7.3 The Tietze extension theorem 153
B. Baire s category theorem
7.4 Nowhere dense sets 157
7.5 Categories and Baire s category theorem 160
7.6 The existence of everywhere continuous nowhere
differentiable functions 165
C. Approximation to continuous functions
7.7 The Weierstrass approximation theorem 170
7.8 Stone s generalization of the Weierstrass
theorem 176
D. Separability
7.9 Separable metric spaces 181
7.10 Examples concerning the separability of metric
spaces 185
APPENDIX
A.1 Properties of metric spaces 190
A.2 Product spaces 191
BIBLIOGRAPHY 194
INDEX 195
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| illustrated | Illustrated |
| indexdate | 2024-07-09T16:57:22Z |
| institution | BVB |
| isbn | 0050024531 |
| language | English |
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| physical | VIII, 198 S. Ill. |
| publishDate | 1972 |
| publishDateSearch | 1972 |
| publishDateSort | 1972 |
| publisher | Oliver & Boyd |
| record_format | marc |
| series | University mathematical texts |
| series2 | University mathematical texts |
| spelling | Pitts, C. G. Verfasser aut Introduction to metric spaces C. G. C. Pitts Edinburgh Oliver & Boyd 1972 VIII, 198 S. Ill. txt rdacontent n rdamedia nc rdacarrier University mathematical texts 42 Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 s DE-604 University mathematical texts 42 (DE-604)BV002783334 42 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004600907&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pitts, C. G. Introduction to metric spaces University mathematical texts Metrischer Raum (DE-588)4169745-5 gnd |
| subject_GND | (DE-588)4169745-5 |
| title | Introduction to metric spaces |
| title_auth | Introduction to metric spaces |
| title_exact_search | Introduction to metric spaces |
| title_full | Introduction to metric spaces C. G. C. Pitts |
| title_fullStr | Introduction to metric spaces C. G. C. Pitts |
| title_full_unstemmed | Introduction to metric spaces C. G. C. Pitts |
| title_short | Introduction to metric spaces |
| title_sort | introduction to metric spaces |
| topic | Metrischer Raum (DE-588)4169745-5 gnd |
| topic_facet | Metrischer Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004600907&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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