MATH 4181 Course Outline

Course Outline for Math 4181 – Introduction to Topology

Course Description: Topics from set theory and point set topology such as car- dinality, order, topological spaces, metric spaces, separation axioms, compactness and connectedness. Based on Textbook: General Topology by Steven Willard (Dover)

Week 1: basics of set-theory, ordered sets, indexed families
Week 2: metric spaces, metric topology, Definition of Topology, closure. Week 3: Neighborhood systems to define topologies, examples, Base for topology, several examples (e.g. Sorgenfrey line) Week 4: separable, first-countable, second-countable, subspace topology Week 5: Continuous maps, Homeomorphisms, embeddings, preservation, Test 1. Week 6: Product topologies, operations in products, quotients Week 7: converging sequences, closure and continuity in first-countable spaces, Separation axioms: T0-T3 Week 8: Tychonoff spaces and normal spaces, Jones’ Lemma for non-normality, behavior in products and subspaces. discuss examples. Week 9: basic covering properties, Uryshohn’s Lemma, Tychonoff’s theorems, Urysohn metrization theorem. Week 10: filter version of compact, Cantor set, local compactness, Lindelof and countable compactness in metric spaces, Tychonoff plank. Week 11: Review and Test 2 Week 12: Complete metric spaces, uniform convergence topology, product of sepa- rable spaces being separable, Baire Category theorem, space filling curve. Week 13: connectedness, path-connected, local versions, components, quasi-components and path-components Week 14: homotopy and Fundamental group (if time) Week 15: Review for Final.