## Metric SpacesThe 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;
- extensive cross-referencing to help the reader follow arguments;
- 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 |

295 | |

297 | |