# Metric Spaces

Springer Science & Business Media, 26.12.2006 - 304 Seiten

The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.

The book goes on to provide a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as: the Tietze Extension Theorem; the Hausdorff metric and its completeness; and the existence of curves of minimum length. Other features include:

• end-of-chapter summaries and numerous exercises to reinforce what has been learnt;
• a Cumulative Reference Chart, showing the dependencies throughout the book on a section-by-section basis as an aid to course design.

The book is designed for third- and fourth-year undergraduates and beginning graduates. Readers should have some practical knowledge of differential and integral calculus and have completed a first course in real analysis. With its many examples, careful illustrations, and full solutions to selected exercises, this book provides a gentle introduction that is ideal for self-study and an excellent preparation for applications.

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### Inhalt

 Metrics 1 11 Metric Spaces Deﬁnition 2 12 Point Functions and Pointlike Functions 8 13 Metric Subspaces and Metric Superspaces 10 14 Isometries 11 15 Extending a Metric Space 12 16 Metrics on Products 13 17 Metrics and Norms on Linear Spaces 16
 93 Uniform Continuity on Subsets of the Cantor Set 153 94 Lipschitz Functions 154 95 Differentiable Lipschitz Functions 156 96 Uniform and Lipschitz Continuity of Compositions 157 97 Uniform and Lipschitz Continuity on Unions 158 98 Uniform and Lipschitz Continuity on Products 159 99 Strong Contractions 160 Summary 162

 Summary 18 EXERCISES 19 Distance 21 22 Distances from Points to Sets 22 23 Inequalities for Distances 24 24 Distances to Unions and Intersections 25 25 Isolated Points 26 26 Accumulation Points 28 27 Distances from Sets to Sets 29 28 Nearest Points 30 Summary 32 EXERCISES 33 Boundary 35 32 Sets with Empty Boundary 37 33 Boundary Inclusion 38 34 Boundaries in Subspaces and Superspaces 41 35 Boundaries of Unions and Intersections 42 36 Closure and Interior 43 37 Inclusion of Closures and Interiors 47 38 Closure and Interior of Unions and Intersections 49 Summary 50 Open Closed and Dense Subsets 53 42 Dense Subsets 57 43 Topologies 58 44 Topologies on Subspaces and Superspaces 61 45 Topologies on Product Spaces 62 46 Universal Openness and Universal Closure 64 47 Nests of Closed Subsets 65 Summary 67 Balls 70 52 Using Balls 75 53 Balls in Subspaces and in Products 77 54 Balls in Normed Linear Spaces 78 Summary 81 Convergence 83 62 Limits 85 63 Superior and Inferior Limits of Real Sequences 86 64 Convergence in Subspaces and Superspaces 88 66 Convergence Criteria for Interior and Closure 90 67 Convergence of Subsequences 91 68 Cauchy Sequences 94 69 Cauchy Sequences in Subspaces 97 611 Forcing Convergence of Cauchy Sequences 99 Summary 100 Bounds 103 72 Finite Products of Bounded Sets 105 74 Spaces of Bounded Functions 107 75 Attainment of Bounds 108 76 Convergence and Boundedness 109 77 Uniform and Pointwise Convergence 110 78 Totally Bounded Sets 113 79 Total Boundedness in Subspaces and Superspaces 116 710 Total Boundedness in Product Spaces 117 711 Solution to the NearestPoint Problem 118 712 Subspaces with the NearestPoint Property 121 Summary 123 Continuity 125 82 Limits of Functions 128 83 Global Continuity 130 84 Open and Closed Mappings 135 85 Continuity of Compositions 136 86 Continuity of Restrictions and Extensions 137 87 Continuity on Unions 138 88 Continuity of Mappings into Product Spaces 139 89 Spaces of Continuous Functions 142 810 Convergence as Continuity 143 Summary 144 EXERCISES 145 Uniform Continuity 147 92 Conservation by Uniformly Continuous Maps 150
 Completeness 165 102 Criteria for Completeness 166 103 Complete Subsets 168 104 Unions and Intersections of Complete Subsets 169 105 Products of Complete Metric Spaces 170 107 Completeness of the Hausdorff Metric 171 108 Complete Spaces of Functions 173 109 Extending Continuous Functions 176 1010 Banachs FixedPoint Theorem 180 1011 Baires Theorem 183 1012 Completion of a Metric Space 187 Summary 188 EXERCISES 189 Connectedness 191 112 Connected Subsets 193 113 Connectedness and Continuity 194 114 Unions Intersections and Products of Connected Sets 195 115 Connected Components 196 116 Totally Disconnected Metric Spaces 198 118 Pathwise Connectedness 200 119 Polygonal Connectedness 202 Summary 204 Compactness 205 122 Compact Subsets 208 123 Compactness and Continuity 209 124 Unions and Intersections of Compact Subsets 211 125 Compactness of Products 212 127 Local Compactness 213 128 Compact Subsets of Function Spaces 215 129 Paths of Minimum Length 217 1210 FiniteDimensional Normed Linear Spaces 220 1211 A Host of Norms 222 Summary 225 Equivalence 227 132 Uniform Equivalence of Metrics 232 133 Lipschitz Equivalence of Metrics 235 134 The Truth about Conserving Metrics 238 136 Equivalent Metric Spaces 240 Summary 243 Language and Logic 245 A2 Truth of Compound Statements 246 A4 Transitivity of Implication 247 A6 Vacuous Truth 248 A8 Proof by Contraposition 249 A10 Existence 250 Sets 251 B2 Subsets and Supersets 252 B3 Universal Set 253 B5 Ordered Pairs and Relations 254 B6 Totally Ordered Sets 255 B7 Extended Real Numbers 257 B8 Ordered Subsets of the Real Numbers 259 B10 Union Intersection and Difference 260 B11 Unions and Intersections of Arbitrary Collections 261 B12 Functions 262 B13 Restrictions Extensions and Compositions 263 B14 Mappings 264 B15 Chains 266 B16 Equivalence Relations 267 B18 Sequences 268 B19 Infinite Selection 269 B20 Algebraic Structures 272 B21 Isomorphism 275 B22 FiniteDimensional Linear Spaces 277 Solutions 279 List of Symbols 292 Bibliography 295 Index 297 Urheberrecht

### Über den Autor (2006)

Mícheál Ó Searcóid is author of another, higher-level, SUMS book, Elements of Abstract Analysis, 1-85233-424-X, published November 2001, sales (as of June 2005): 1051 (ROW: 634; US: 417).