## Analytical topology |

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### Contents

SETS FUNCTIONS AND ORDERINGS | 3 |

BASIC CONCEPTS FOR METRIC SPACES | 21 |

TYPES OF METRIC SPACE | 72 |

Copyright | |

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### Common terms and phrases

arbitrary non-null axiom belongs clearly closed subsets cluster point compact space compact subset complete lattice completely separable connected subset contains Conversely Corollary countably compact defined DEFINITION denote the set dense dense-in-itself discrete discrete topology disjoint equivalent Example Exercises for Solution family of members finite intersection property finite sequence finite subset follows from Theorem Fr(X G n G Given a non-null Given a space Given an arbitrary Given spaces S,p hence homeomorphic property implies infimum infinite lattice locally compact member G metric space Ne(x non-enumerable non-null set non-null subset Note one-point compactification open covering open sets open subset p,p')-continuous poset positive integer product space r-base r-closed real numbers resp satisfies condition Secondly sequence of elements sequence of members sequence xj T2-space Theorem Theorem 2.5 topological space S,r topology totally bounded trivial topology Tukey's Lemma usual metric x e G