## 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 disjoint equivalent Example family of members finite intersection property finite sequence follows from Theorem Fr(X Given a space Given an arbitrary given non-null Given spaces S,p hence homeomorphic property implies infimum infinite subset jV(x jV\x lattice locally compact member G metric space Ne(x neighbourhood non-enumerable non-null set non-null subset notation Note null one-point compactification open covering open sets open subset p,p')-continuous poset positive integer product space real numbers resp satisfies condition Secondly sequence of elements sequence of members sequence xj subbase subset of P(S t-base t-closed t-open sets T2-space Theorem Theorem 2.5 topological space S,t topology totally bounded trivial topology Tukey's Lemma usual metric x e G XuX2