## MAT 327: Introduction to Topology, Fall 2012

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

TA: Sasha Pavlov; sasha dot pavlov at utoronto dot ca

Lectures: Mondays 2-3pm, Thursdays 2-4pm at SS 1086
On Wednesday, Dec 5, 2-3pm we'll have a make-up class. Location: SS 1086 (usual classroom)

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)

• Homework (roughly 8 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 25, 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.

### Homework

• Assignment 1 due Thu Sep 27: section 13, problems 4, 7 (only T1, T2, T3), 8; section 16, problems 4, 8
(Note that in the last problem the answer may depend on the slope of the line.)
• Assignment 2 due Thu Oct 4: section 17, problems 6, 11, 13, 16 (only topologies T2, T4, and T5), 18 (only A, D, E); section 18, problem 13
(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 11: section 18, problems 2, 4, 5, 8, 9ab (compare this with thm 18.2(f) which we didn't discuss in class), 11; section 19, problems 6, 7
• Assignment 4 due Mon Oct 22 at 3pm: section 19/8, section 20/ 1, 3, 4a* (only f), 4b* (only x,y,z), 5*, 6, 7, 8a*, 8b, section 21/ 1, 2, 3a*, 3b
Only the problems marked with a * will be collected! (i.e., 4a, 4b, 5, ...)
• Assignment 5 due Thu Nov 8 at 4pm: section 23/ 2*, 3, 4, 7, 9*, section 24/ 1*, 2, 10*, section 26/ 1, 2*, 3, 4*, 5*, 7
(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.)
[the problems that aren't starred are meant for practice, you don't have to do them]
• Assignment 6 due Mon Nov 19 at 3pm: section 27/ 2*, section 28/ 1, 2*, 4*, 6, 7a*, 7b*, 7c*, section 31/ 4*
(for 27/2, see definition on p. 175)
• Assignment 7 due Mon Nov 26 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).)
for 35/9, please note that it should say "a set U is open in X if U intersect Xi is open in Xi for each i" (an "i" is missing in some versions of the text book)
• Assignment 8 due Wed Dec 5 at 3pm: section 35/ 4*, 5, section 37/ 1* [Note: in part (c) there is a typo: you need to assume T2!], section 51/ 1*, 2*, 3*
Exercise (optional): 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 25, 2:10-4:00pm
• Location: MP 202
• The test will cover all material up to and including Oct 18
• Last year's term test
• Term test 2010 (Comment: please interpret "stronger" in Q3 as "finer".)

### Final

• Thu, Dec 13, 2-5pm
• Location: BN3
• Last year's final
• Final from 2010 (Comment: we didn't discuss the material relevant to questions 4 and 6.)
• Office Hours before the final: TBA