## MAT 327: Introduction to Topology, Fall 2011

Instructor: Florian Herzig; my last name at math dot toronto dot edu
Office Hours: Wednesdays 10:30am-12:30pm at BA 6186

TA: Micheal Pawliuk; m dot pawliuk at utoronto dot ca

Lectures: Mondays 2-3pm, Thursdays 2-4pm at SS 1086
On Monday, Dec 5, we'll have a make-up class 3-4pm (just after the regular class time). Location: MP 118 (McLennan Physical Labs)

Class photo

### Topics to be covered (rough plan)

• topological spaces, continuous functions, product and subspace topology
• metric topology, connectedness, compactness
• Hausdorff/regular/normal spaces, Urysohn's lemma, Tietze extension theorem
• Tychonoff's theorem
• fundamental group, Brouwer fixed-point theorem

### Textbook

• James Munkres, Topology, 2nd ed.
Also recommended: Jänich Topology (for intuition and general picture), Armstrong Basic Topology (especially for last part on homotopy)

### Grading scheme

• Homework (roughly 9 assignments): 25%
• Term test: 25%
• Final: 50%
Homeworks will be usually be posted on this web page on Thursdays and be due on the following Thursday (in class or by e-mail to the TA). No late homework will be accepted! Your lowest two homework scores will not count towards your grade.

The term test will be on Thu, October 27, 2-4pm. There will be no makeup test! If you miss the test for a valid reason, the grade will be reweighted as 35% homework and 65% final.

The final will be on Thu, December 15, 9am-12pm at SS 2127.

### Homework

• Assignment 1 due Thu Sep 29: section 13, problems 4, 6, 8; section 16, problems 4, 5, 8
(Note that in the last problem the answer may depend on the direction of the line.)
(Note on problem 5: you can think of X and X' as one set, let's call it X_0, that has two different topologies, T and T'; so X = (X_0, T) and X' = (X_0, T'). Similarly for Y and Y'. I hope that's clearer!)
• Assignment 2 due Thu Oct 6: section 17, problems 6, 13, 16, 17, 19, 20; section 18, problem 13
(You don't have to give very detailed arguments for problem 20, just make clear you understand what's going on.)
(Problem 21 in section 17 is somewhat tricky but fun, if you want to have something else to think about. It won't be collected.)
• Assignment 3 due Thu Oct 13: section 18, problems 4, 5, 8, 9ab; section 19, problems 5 (say why), 6, 7, 8
• Assignment 4 due Mon Oct 24 at 3pm:
Part 1:
section 20/ 1*, 4a, 4b*, 5, 6, 8a*, 8b* section 21/ 2, 3a*, section 23/ 4, 5*, 7* (other good practice problems: 20/ 3, 7; 21/ 1, 3b; 23/ 1)
Only the problems marked with a * will be collected! (i.e., 1, 4b, 8a, ...)
Part 2: send me your description in the class photo, if you haven't already; send me a photo of yourself if you missed the class photo
• Assignment 5 due Thu Nov 10 at 4pm: section 23/ 2*, section 24/ 1*, 2, 10*, section 26/ 2*, 3*, 5*, 7 (other good practice problems: 23/10, 26/1; more challenging: 26/ 8, 9, 11)
(Hint for 26/7: use the tube lemma. Comment on 24/10: the pasting lemma could be useful to justify one step in the argument.)
• Assignment 6 due Mon Nov 21 at 3pm: section 27/ 2*, section 28/ 1, 2, 4*, 6, 7a*, 7b*, 7c*, section 31/ 4*, 6*
(for 27/2, see definition on p. 175)
• Assignment 7 due Mon Nov 28 at 3pm: section 31/ 2*, section 32/ 1*, section 33/ 3, 4*, 5*, 8, section 35/ 1, 9*
(see def. of d(x,A) on p. 175, def. of completely regular on p. 211; hint for 33/4: it might help to construct f as infinite series sum(f_n).)
• Assignment 8 due Wed Dec 7 at 5pm: section 35/ 4*, section 37/ 1a*, 1b, 1c* [Note: in part (c) there is a typo: you need to assume T2!], section 51/ 1*, 2, 3*
Exercise*: Show that {0,1}^omega is homeomorphic to the Cantor set without using Tychonoff's theorem. Deduce that {0,1}^omega is compact.

### Term test

• Thu Oct 27, 2:10-4:00pm
• Location: EX 320
• The test will cover all material up to Oct 20 (connectedness)
• Last year's term test (Comment: please interpret "stronger" in Q3 as "finer".)
• Office Hours in the week of Oct 24-28: Wed Oct 26, 5:30-7:30pm (or by appointment)
Micheal will also e-mail you about his office hours (Tue Oct 25, 10-11am, Wed Oct 26, 11-12pm at BA 6283)

### Final

• Thu, Dec 15, 9am-12pm
• Location: SS 2127
• Last year's final (Comment: we didn't discuss the material relevant to questions 4 and 6.)
• Office Hours before the final: Tue Dec 6, 1:30-3pm, Mon Dec 12, 2-3:30pm, Tue Dec 13, 3:30-5pm, Wed Dec 14, 5-6:30pm
Micheal's office hours (Mon/Tue 6:10-8:00) are posted on Blackboard

### Detailed schedule and lecture notes

Thanks to Hayoon for providing lecture notes!

Day Topics Book sections Notes
9/12 introduction   pdf
9/15 topological spaces, continuous maps, bases 12, 13 pdf
9/19 more on bases, subbases 13 pdf
9/22 subspace and product topology 15, 16 pdf
9/26 order topology, closed sets, closure 14, 17 pdf
9/29 limit points, some separation axioms (Hausdorff = T2, T1), more on continuity 17, 18 pdf
10/3 more on continuity, homeomorphisms 18 pdf
10/6 product topology in general 19 pdf
10/10 (Thanksgiving, no class)
10/13 metric topology 20, 21 pdf
10/17 connectedness 23, 24 pdf
10/20 more on connectedness, path connectedness 23, 24 pdf
10/24 compactness 26, 27 pdf
10/27 Term Test
10/31 more on compactness 26, 27 pdf
11/3 compactness in metric spaces 27, 28, 45 pdf
11/7 (Fall break, no class)
11/10 compactness (end), the Cantor set 27, 28, 45 pdf
11/14 separation axioms: T3, T4 31, 32 pdf
11/17 more on separation axioms, Urysohn's lemma 31, 32, 33 pdf*
11/21 Tietze extension theorem 35 pdf
11/24 Tychonoff's theorem: Micheal's notes 37 pdf
11/28 homotopy of paths 51 pdf
12/1 more on homotopy, fundamental group, case of circle (beginning) 51, 52, 53, 54 pdf*
12/5 fundamental group of circle (conclusion), Brouwer fixed point and Borsuk-Ulam 54, 55, 57 pdf

*These two PDF files only seem to work with apple's preview. Hayoon provided alternative versions: pdf pdf