MAT 327: Introduction to Topology, Fall 2011

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.

Grading scheme

• Homework (roughly 9 assignments): 25%
• Term test: 25%
• Final: 50%

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

Links

• Last year's course web page
• Errata in Munkres book
• Any compact metric space is a continuous image of the Cantor set This link has some typos, e.g. I^{infinity} = I^{natural numbers}, but overall it seems fine.
• Tychonoff's theorem implies the axiom of choice
• Youtube video illustrating path lifting
• Proof of the fundamental theorem of algebra in Hatcher's book (see theorem 1.8)