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 R∞: (a) `1, `2; `∞; (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, 1} 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.