## Math 342: Topology, Spring 2016

MWF 1:10-2pm, Library 204
Office Hours: Mon 11-12, Wed 2-3, and by appointment in Library 313

Textbook: *Topology*, 2nd ed., by James Munkres.

### Week 1: January 25-29

- Monday: Course overview. Hand out syllabus.
- Wednesday: Set theory review.
- Friday: Read §12. Topological spaces.

#### Homework due Friday

- Review the syllabus.
- §1: 5, 8, 9.
- §2: 2.
- §5: 1.

### Week 2: February 1-5

- Monday: Read §13. Bases and subbases for topologies.
- Wednesday: Read §18 (skip portions involving subspaces or products). Continuous functions.
- Friday: Continuity and categories.

#### Homework due Friday

- §13: 1, 3, 4, 5, 8
- §17: 4
- Do the problems in this handout [pdf].

### Week 3: February 8-12

- Monday: Read §§1-2 of these notes [pdf]. Entry quiz. Categories, isomorphisms, and homeomorphisms.
- Wednesday: Read §15 and §3 of cat.pdf. The product topology.
- Friday: Read §16 and §4 of cat.pdf. The subspace topology.

#### Homework due Friday

- §18: 5, 6, 7, 9 (you will need to use Theorem 18.3)
- Do the problems in this handout [pdf].

### Week 4: February 15-19

- Monday: Read §17. Hausdorff spaces.
- Wednesday: Review session based on these questions [pdf].
- Friday: In-class exam. Four questions, 50 minutes, one question drawn from the review problems. You may bring one two-sided 8.5"x11" cheat sheet, but no other resources are permitted.

#### No homework this week.

### Week 5: February 22-26

- Monday: Review §17. Clean up closed sets, limit points, and Hausdorff spaces. Talk about exam.
- Wednesday: Read §19 in Munkres and §5 in cat.pdf. Infinite products.
- Friday: Review Wednesday's reading assignment. HW due. Product vs. box topology cage match showdown.

#### Homework due Friday

- §17: 9, 13, 14, 17, 21 (bonus)
- §18: 12
- §19: 4, 8

### Week 6: February 29 - March 4

- Monday: Read §22 in Munkres and §6 in cat.pdf. Entry quiz. The quotient topology.
- Wednesday: Read §20 in Munkres. Metric spaces I.
- Friday: Read §21 in Munkres. Metric spaces II.

#### Homework due Friday

- §19: 2, 7, 10
- §22: 2, 3, 4
- §20: 1

### Week 7: March 7-11

- Monday: Read §23. Entry quiz. Metric spaces III. Connected spaces.
- Wednesday: Read §24. Connected subspaces of $\mathbb{R}$.
- Friday: Read §25. Components and local connectedness.

#### Homework due Friday

- §20: 4, 5, 8 (you may assume the results of Exercise 10)
- §21: 2, 6
- A function between metric spaces $f:X\to Y$ is called
*distance-decreasing*(or*dd*) if $d(f(x),f(x'))\le d(x,x')$ for all $x,x'\in X$. (Here we allow $d$ to denote the metric on both $X$ and $Y$.) - Show that all
*dd*functions are continuous. - Prove that the category $\mathscr{M}_{dd}$ of metric spaces
and
*dd*functions is in fact a category. - Prove that a function between metric spaces is an isomorphism in $\mathscr{M}_{dd}$ if and only if it is a surjective isometry (cf. Exercise 21.2 from Munkres).

### Week 8: March 14-18

- Monday: Read §25. Entry quiz. Components and local connectedness.
- Wednesday: Read §26. Compact spaces.
- Friday: Read §27. Compact subspaces of $\mathbb{R}$. Talk about topic proposals [pdf] for end-of-term presentations.

#### Homework due Friday

- §23: 1, 2, 5, 7, 11
- §24: 1, 3, 5, 9

### Spring Break: March 21-25

### Week 9: March 28 - April 1

*Announcement*: Amazing topology talks upcoming at Reed: Jeremiah Heller on vector bundles (April 1), Steve Awodey on homotopy type theory (April 7), Dan Dugger on elliptic curves and Pascal's theorem (April 14), Vesna Stojanoska on shapes of numbers (April 21), and Haynes Miller on knots and numbers (April 28)!

- Monday: Topic proposal due. Distribute take-home exam (due Friday). Extreme value theorem. Uniform continuity. Exam review.
- Wednesday: Read §51. Path homotopies.
- Friday: Turn in take-home exam. Read §52. The fundamental group(oid).

#### Take-home exam due at the start of class on Friday.

### Week 10: April 4-8

- Monday: Entry quiz. Read §7 of cat.pdf. Functoriality of $\pi_1$.
- Wednesday: Read §53. Covering spaces
- Friday: Read §54. More covering spaces and $\pi_1(S^1)$.

#### Homework due Friday

- §51: 1, 2, 3
- §52: 3, 4, 5
- Show that $\pi_0$ is a functor from Top to Set. (Show that the morphism assignment is well-defined and prove that the other functor properties hold.)
- Define the category of groupoids, Gpoid.
- Prove that $\Pi_1$ is a functor from Top to Gpoid.

### Week 11: April 11-15

- Monday: Read §55. Entry quiz. The Brouwer fixed point theorem.
- Wednesday: Read §57. The Borsuk-Ulam theorem.
- Friday: Read §58. Deformation retracts, homotopy type, and the homotopy category.

#### Homework due Friday

- §53: 3, 5, 6
- §54: 2, 3, 4, 6
- §55: 4(a,b,c) [You are encouraged to think about parts (d,e,f) as well, but do not need to write them up.]

### Week 12: April 18-22

- Monday: Read §§68-69. Seifert-van Kampen I.
- Wednesday: Read §70. Seifert-van Kampen II.
- Friday: Student presentations. Taylor Allen,
*Dimension theory and embeddings of manifolds*; Sara Jo Weinstein,*Stone-Čech compactification*.

#### Homework due Friday

*Note:* Students presenting on Friday get an automatic extension until
Monday, April 25.

- §55: 2
- §57: 2
- §58: 9(a,b,c,d)

Problems 9(e) and 10 from §58 are highly recommended, but also completely optional.

### Week 13: April 24-29

- Monday: Student presentations. Sam Johnston,
*Stone representation theorem of Boolean algebras*; Ben Schloesser,*Distributive lattices and coherent spaces*. - Wednesday: Student presentations. Forrest Glebe,
*(When it's meaningful to say) "You're so far away from me" (metrizability)*; Nico Terry,*Countability, separation, and COOL EXAMPLES*. - Friday: Student presentations. Tanmay Dubey,
*Higher homotopy groups*; Palak Jain,*Topological groups*. Take-home final distributed. Due May 6 by 5pm.

#### Homework due Friday

- None.

### The $\LaTeX$ document preparation system

Poor handwriting? Love escape characters? Too much free time? Try $\LaTeX$!

- $\LaTeX$ at Reed.
- A short guide [pdf] to writing mathematics with $\LaTeX$.
- Sample $\LaTeX$ input / output: sample.tex / sample.pdf.

Kyle M. Ormsby