MATH 131 (2017), DAY-BY-DAY CONTENT Week 1. Day 1, Thurs., Aug 31: 1) Metric spaces: (a) Recall the notion of continuous map Rn ! R; (b) The notion of metric space; (c) Examples; (d) Continuity for maps of metric spaces. 2) Open subsets in a metric space: (a) The notion of open subset; (b) Continuity via open subsets. 3) Topological spaces: (a) The notion of topological space; (b) Examples. Week 2. Day 1, Tue., Sept 5: 1) Continuous maps between topological spaces: (a) Definition of continuous maps; (b) Maps from discrete; (c) Maps to trivial; (d) Maps out of disjoint union. 2) Product topology: (a) Basic for a topology; (b) Continuity via a basis; (c) Definition of the product topology; (d) Definition by a universal property. 3) Separation axioms: (a) T1 axiom, counterexamples; (b) T2 (Hausdorff), Zariski topology. Date: November 21, 2017. 1 2 MATH 131 (2017), DAY-BY-DAY CONTENT Day 2, Thurs., Sept 7: 1) Homeomorphisms: (a) Definition; (b) Examples. 2) Compactness: (a) Definition; (b) Heine-Borel (the unit interval is compact); (c) Image of compact in Hausdorff is closed. Week 3. Day 1, Tue., Sept 12: 1) 1st countability: (a) Definition; (b) Closedness via sequential closedness. 2) 2nd countability: (a) Definition; (b) Examples; (c) Compatness vs sequential compactness. Day 2, Thurs., Sept 14: 1) Compactness for metric spaces: (a) Compatness=sequential compactness; (b) Compatness=complete+totally bounded. 2) Topologies on R1: (a) `1, `2; `1; (b) Non-compactness of the cube in any of the above; (c) The Hilbert cube. 3) Infinite products (introduction): (a) Introduce topology on the infinite product; (b) Mention universal property; (c) State Tychonoff; (d) Topology on the Hilbert cube. MATH 131 (2017), DAY-BY-DAY CONTENT 3 Week 4. Day 1, Tue., Sept 19: 1) Topology on an infinite product: (a) Definition, mention universal property; (b) Hilbert cube; (c) State Tychonoff's theorem. 2) Proof of Tychonoff: (a) Compactness via pre-base; (b) Check pre-base compactness condition; (c) Zorn's lemma (proof for countable); (d) Proof of compactness via pre-base. Day 2, Thurs., Sept 21: 1) The space of continuous functions from a compact to a metric: (a) Definition; (b) Why we need X compact; (c) Completeness; (d) Non-compactness. 2) Equicontinuity: (a) Definition; (b) Continuous function on a compact is equicontinuous. 3) Construction of the intergal: (a) Piece-wise continuous and constant functions; (b) Integral as a functional on piece-wise constant functions; (c) Density of piece-wise constant in piece-wise continuous via equicontinuity. Week 5. Review 4 MATH 131 (2017), DAY-BY-DAY CONTENT Week 6. Day 1, Tue., Oct. 3: 1) Normal spaces: (a) Definition; (b) Compact + Hausdorff ) normal; (c) Metric spaces are normal. 2) Urysohn's lemma and metrization: (a) Urysohn's lemma; (b) Metrization for normal and 2nd countable: statement; (c) Proof via embedding into Hilbert cube. Day 2, Tue., Oct. 5: 1) Locally compact spaces: (a) Definition; (b) 1-point compactification: construction; (c) 1-point compactification: Hausdorff; (d) 1-point compactification: universal property for mapping in. 2) Compact-open topology: (a) Reminder: topology of uniform convergence on C(X; Y ); (b) Definition of compact-open topology; (c) Comparison with uniform for X compact; (d) Continuity of the evaluation map. MATH 131 (2017), DAY-BY-DAY CONTENT 5 Week 7. Day 1, Tues., Oct. 10: 1) Connectedness: (a) Definition; (b) [0,1] is connected; (c) Image of connected is connected; (d) Intermediate value theorem. 2) Path-connectedness: (a) Definition; (b) sin counter-example; (c) Decomposition of locally path-connected into path-connected components; (d) Equivalence of connectedness and path-connected for locally path-connected, example of domains in Rn. 3) Decomposition into connected components: (a) Construction of decomposition; (b) Totally disconnected spaces; (c) Q is totally disconnected. Day 2, Thurs., Oct. 12: 1) The notion of homotopy: (a) Definition; (b) Equivalence relation; (c) Homotopy as a path in the space of maps. 2) Homotopy equivalences: (a) Definition; (b) Contraction of Rn to pt; (c) Contraction of Rn − 0 to Sn−1, (d) Contraction of R2 − {−1; 1g to figure 8. 3) The fundamental group: (a) Loops and homotopies of loops; (b) The operation of concatenation; (c) Unit for the group law. 6 MATH 131 (2017), DAY-BY-DAY CONTENT Week 8. Day 1, Tues., Oct. 17: 1) The fundamental group, continued: (a) Recap: the multiplication; (b) Inverse; (c) Associativity. 2) Properties: (a) Functoriality for maps of spaces; (b) Change of base point; (c) Behavior under homotopies. Day 2, Thurs., Oct. 19: 1) Covering spaces: (a) Definition; (b) Example of R ! S1; (c) Homotopy lifting property (with proof for pt). 2) Action of paths on fibers of a covering space: (a) Moving between fibers along a path; (b) Independence up to homotopy; (c) Action of π1 on the fiber. (d) Identification of the stabilizer. MATH 131 (2017), DAY-BY-DAY CONTENT 7 Week 9. Day 1, Tues., Oct. 24: 1) Recap: (a) Homotopy lifting property; (b) Lifting of paths and identification of fibers. 2) Covering spaces and the action of π1 on the fiber: (a) Connectedness via transitivity; (b) Detect isomorphismsms; (c) Identify stabilizer of a point. Day 2, Thurs., Oct. 26: 1) Breaking a covering space into connected components: (a) Connected components are covering; (b) Orbits on the fiber and connected components. 2) Fully faithfulness of the functor of fiber: (a) Injectivity; (b) Interlude: fiber products; (c) Proof of surjectivity. Week 10. Day 1, Tues., Oct. 31: 1) Existence of covers corresponding to a given set with an action of π1(X; x). Day 2, Tues., Nov 2: Review 8 MATH 131 (2017), DAY-BY-DAY CONTENT Week 11. Day 1, Tues., Nov. 7: No class{was sick. Day 2, Tues., Nov. 9: 1) Solve midterm. 2) Applications of the fundamental group: (a) No retraction of disc on circle; (b) Fixed points on 2-ball; (c) Fundamental theorem of algebra. 3) Subgroups and covering spaces. 4) Proof of homotopy lifting property (just for paths). Week 12. Day 1, Tues., Nov. 14: 1) Proof of the existence of the universal cover. Day 2, Tues., Nov. 16: 1) The Klein bottle: (a) Construction as a quotient of a torus; (b) Semi-direct products; (c) Covering by R2 and description of the fundamental group; (d) Description of subcovers (incl. Moebius strip). 2) Push-outs of groups: (a) Products and coproducts in various categories; (b) Coproducts of groups; (c) Push-outs of groups. MATH 131 (2017), DAY-BY-DAY CONTENT 9 Week 13. Day 1, Tues., Nov. 21: 1) Van Kampen theorem: (a) Recall push-outs of groups; (b) Statement of Van Kampen; (c) Pushouts of topological spaces along open embeddings. 2) Applications of Van Kampen: (a) Simply-connectedness of S2; (b) n-punctured plane; (c) Genus n surfaces with one puncture; (d) Non-punctured genus n surfaces; (e) Genus n surfaces with multiple punctures. 3) Proof of Van Kampen via covering spaces: (a) Universal property of push-out via actions; (b) Reformulation of VK via covering spaces; (c) Proof via gluing of covering spaces. Day 2, Tues., Nov. 23: Thanksgiving break..