## Topology of Metric Spaces"Topology of Metric Spaces 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, to treat this as a preparatory ground for a general topology course, to use this course as a surrogate for real analysis and to help the students gain some perspective of modern analysis." "Eminently suitable for self-study, this book may also be used as a supplementary text for courses in general (or point-set) topology so that students will acquire a lot of concrete examples of spaces and maps."--BOOK JACKET. |

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P126 has a flaw,which is lack of a '<'between two sigma.

### Contents

Basic Notions | 1 |

Convergence | 35 |

Continuity | 56 |

Compactness | 81 |

Connectedness | 106 |

Complete Metric Spaces | 122 |

Bibliography | 149 |

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### Common terms and phrases

admit a finite arbitrary topological space Archimedean property Assume bijection Cauchy sequence choose claim closed sets closed subset cluster point compact metric space compact subset completes the proof conclude connected subset constant contradiction convergent sequence convergent subsequence countable d(xm,xn dA(x define Definition denote dense sets dense subset disjoint equivalent Example Figure finite subcover function g geometric given graph Hausdorff Hint homeomorphic iff there exists induced metric infinite intersection isometry last exercise Lemma Let X,d Let xn limit point Lipschitz nonempty set nonzero notation open ball open cover open intervals open set open subset path connected product metric Proposition prove reader real numbers Remark result Rn+i sequence xn standard metric subsets of R2 sup norm topological property topological space totally bounded triangle inequality uniformly continuous uniformly equicontinuous union upper bound vector space vector subspace