Contents
4 Topological Spaces 2
4.1 Topological Spaces: Denitions and Examples . . . . . . . . . 2
4.2 Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.3 Subspace Topologies . . . . . . . . . . . . . . . . . . . . . . . 4
4.4 Continuous Functions between Topological Spaces . . . . . . . 6
4.5 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.6 Sequences and Convergence . . . . . . . . . . . . . . . . . . . 8
4.7 Neighbourhoods, Closures and Interiors . . . . . . . . . . . . . 9
5 Product Topologies 10
5.1 The Cartesian Product of Subsets of Euclidean Space . . . . . 10
5.2 Product Topologies . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Identication Maps and Quotient Topologies 18
6.1 Cut and Paste Constructions . . . . . . . . . . . . . . . . . . . 18
6.2 Identication Maps and Quotient Topologies . . . . . . . . . . 20
7 Compactness 23
7.1 Compact Topological Spaces . . . . . . . . . . . . . . . . . . . 23
7.2 Compact Metric Spaces . . . . . . . . . . . . . . . . . . . . . . 29
7.3 The Lebesgue Lemma and Uniform Continuity . . . . . . . . . 33
8 Connectedness 34
8.1 Connected Topological Spaces . . . . . . . . . . . . . . . . . . 34