Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres' textbook John Rognes November 21st 2018 Contents Introduction v 1 Set Theory and Logic 1 1.1 (x1) Fundamental Concepts . 1 1.1.1 Membership . 1 1.1.2 Inclusion and equality . 2 1.1.3 Intersection and union . 2 1.1.4 Difference and complement . 2 1.1.5 Collections of sets, the power set . 3 1.1.6 Arbitrary intersections and unions . 3 1.1.7 Cartesian products . 4 1.2 (x2) Functions . 4 1.2.1 Domain, range and graph . 4 1.2.2 Image, restriction, corestriction . 5 1.2.3 Injective, surjective, bijective . 5 1.2.4 Composition . 6 1.2.5 Images of subsets . 6 1.2.6 Preimages of subsets . 7 1.3 (x5) Cartesian Products . 8 1.3.1 Indexed families . 8 1.3.2 General intersections and unions . 8 1.3.3 Finite cartesian products . 9 1.3.4 Countable cartesian products . 9 1.3.5 General cartesian products . 10 1.4 (x6) Finite Sets . 10 1.4.1 Cardinality . 10 1.4.2 Subsets . 11 1.4.3 Injections and surjections . 12 2 Topological Spaces and Continuous Functions 13 2.1 (x12) Topological Spaces . 13 2.1.1 Open sets . 13 2.1.2 Discrete and trivial topologies . 13 2.1.3 Finite topological spaces . 14 2.1.4 The cofinite topology . 15 2.1.5 Coarser and finer topologies . 16 2.1.6 Metric spaces . 17 2.2 (x13) Basis for a Topology . 18 2.2.1 Bases . 18 i 2.2.2 Comparing topologies using bases . 21 2.2.3 Subbases . 22 2.3 (x15) The Product Topology on X × Y ........................ 22 2.3.1 A basis for the product topology . 22 2.3.2 A subbasis for the product topology . 23 2.4 (x16) The Subspace Topology . 24 2.4.1 Subspaces . 24 2.4.2 Products vs. subspaces . 26 2.5 (x17) Closed Sets and Limit Points . 26 2.5.1 Closed subsets . 26 2.5.2 Closure and interior . 28 2.5.3 Closure in subspaces . 29 2.5.4 Neighborhoods . 29 2.5.5 Limit points . 30 2.5.6 Convergence to a limit . 30 2.5.7 Hausdorff spaces . 31 2.5.8 Uniqueness of limits in Hausdorff spaces . 31 2.5.9 Closed sets and limits points in Hausdorff spaces . 31 2.5.10 Products of Hausdorff spaces . 32 2.6 (x18) Continuous Functions . 32 2.6.1 Continuity in terms of preimages . 32 2.6.2 Continuity at a point . 33 2.6.3 Continuity in terms of closed sets and the closure . 33 2.6.4 Homeomorphism = topological equivalence . 34 2.6.5 Examples . 35 2.6.6 Nonexamples . 37 2.6.7 Constructing maps . 37 2.6.8 Maps into products . 38 2.6.9 Maps out of products . 39 2.7 (x19) The Product Topology . 41 2.7.1 Pointwise convergence . 43 2.7.2 Properties of general product spaces . 44 2.8 (x20) The Metric Topology . 44 2.8.1 Bounded metrics . 44 2.8.2 Euclidean n-space . 45 2.8.3 Infinite dimensional Euclidean space . 45 2.9 (x21) The Metric Topology (continued) . 47 2.10 (x22) The Quotient Topology . 48 2.10.1 Equivalence relations . 48 2.10.2 Quotient maps . 49 2.10.3 Open and closed maps . 50 3 Connectedness and Compactness 55 3.1 (x23) Connected Spaces . 55 3.1.1 Sums of spaces . 55 3.1.2 Separations . 55 3.1.3 Constructions with connected spaces . 56 3.2 (x24) Connected Subspaces of the Real Line . 58 3.2.1 Path connected spaces . 59 ii 3.3 (x25) Components and Local Connectedness . 60 3.3.1 Path components . 61 3.3.2 Locally connected spaces . 61 3.3.3 The Jordan curve theorem . 62 3.3.4 Gaussian elimination . 62 3.4 (x26) Compact Spaces . 64 3.4.1 Open covers and finite subcovers . 64 3.4.2 Compact subspaces of Hausdorff spaces . 66 3.4.3 Finite products of compact spaces . 67 3.4.4 The finite intersection property . 68 3.5 (x27) Compact Subspaces of the Real Line . 69 3.5.1 The Lebesgue number . 70 3.5.2 Uniform continuity . 71 3.5.3 The Gram{Schmidt process . 71 3.6 (x28) Limit Point Compactness . 72 3.7 (x29) Local Compactness . 74 3.7.1 The one-point compactification . 75 3.7.2 The local nature of local compactness . 77 4 Countability and Separation Axioms 78 4.1 (x30) The Countability Axioms . 78 4.1.1 First-countable spaces . 79 4.1.2 Second-countable spaces . 80 4.1.3 Countable dense subsets . 80 4.2 (x31) The Separation Axioms . 81 4.3 (x32) (More About) Normal Spaces . 83 4.4 (x33) The Urysohn Lemma . 84 4.5 The Hilbert Cube . ..