## Metric SpacesEncouraged by the response to the first edition the authors have thoroughly revised Metric Spaces by incorporating suggestions received from the readers. |

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### Contents

V | 1 |

VII | 3 |

VIII | 4 |

IX | 5 |

XI | 6 |

XII | 9 |

XIV | 16 |

XV | 19 |

XXXIV | 74 |

XXXV | 79 |

XXXVII | 85 |

XXXVIII | 90 |

XXXIX | 92 |

XL | 95 |

XLII | 98 |

XLIII | 102 |

### Common terms and phrases

aeA aeA arbitrary bijective bounded sets Cantor set Cauchy sequence closed and bounded closed interval closed sets closed sphere closed subset complete metric space contains continuous function convergent sequence convergent subsequence Corollary d(xm d(xn Definition A metric Definition Let denoted dense set disconnected discrete metric space disjoint elements equivalent metrics Example finite number finite subcover follows G is open given Hence homeomorphism interior point intersection isometry Let x e lim xn limit point mapping metric space Ru non-empty open set non-empty sets Note open cover open interval open set open sphere centred open subsets positive integer Problem Proof Let Prove real numbers separated sets sequence xn sequentially compact space and let space Xd subspace Theorem Let totally bounded triangle inequality uncountable set uniformly continuous uniformly equivalent union usual metric space verifies view of Theorem