## Topology: An IntroductionThis book provides a concise introduction to topology and is necessary for courses in differential geometry, functional analysis, algebraic topology, etc. Topology is a fundamental tool in most branches of pure mathematics and is also omnipresent in more applied parts of mathematics. Therefore students will need fundamental topological notions already at an early stage in their bachelor programs. While there are already many excellent monographs on general topology, most of them are too large for a first bachelor course. |

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### Contents

2 | |

5 | |

3 Construction of Topological Spaces | 41 |

4 Convergence in Topological Spaces | 59 |

5 Compactness | 73 |

6 Continuous Functions | 87 |

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algebra Analogously arbitrary assume Axiom Baire space Cantor set Cartesian product characterization closed subset closure cluster point compact Hausdorff space compact spaces compact subsets connectedness Consider contains continuous functions continuous map Corollary countably compact define definition dense subsets disjoint equicontinuous Example Exercise f is continuous filter F finer finite subcover function f Hausdorff space Hence homeomorphism index set Lemma Let f Let M,M locally compact Hausdorff locally uniform convergence map f meager subset metric space Moreover neighbourhood basis notions numbers open balls open cover open interior open neighbourhood open subsets ºſ(p path-connected pi)ieſ pointwise preimage product topology Proof Let Proposition 7.1.5 respect second countable seminorms separation properties sequence Show space and let Springer subalgebra subbasis subset A C M subspace topology topological manifolds topological space topological space M,M totally bounded ultrafilter union