Here is a comment on test 7. The Baby Van Kampen is Theorem 59.1 in Munkres. The last homework problem, for Monday: Let X and Y be spaces with the basepoints x and y. Let (x,y) be the basepoint of the product X x Y. Prove that the fundamental group of X x Y is the direct product of the fundamental groups of X and Y. Here are the end-of-term exam problems and (coming soon) answers. And this is a copy of test 7. As I just emailed you, problem 3 in this test has been withdrawn. Monday, December 1: ## 3,5,7 from section 54 (#3 is about the end of the computation today) Here is a comment on test 6. Monday, November 24: read section 53, especially the proofs that I skipped and Example 2. Do #5 on page 341. The end-of-term exam is next Wednesday, in class. It will be on the material from chapter 3 and chapter 4. Again, there will be two proofs and several yes/no questions. The last take home test given out next week will be entirely on the fundamental group and covering spaces. Friday, November 21: ## 4,5,6 from section 52. Wednesday, November 19: ## 1,2,3 from section 52. I noticed that when I put a bar over a path, Munkres puts a hat. These mean the same thing---the same path traveled in reverse. Monday, November 17: no new homework problems but here is an advance copy of test 6. Friday, November 14: you can read section 51 to the end of the examples on page 325 and do ## 1,2,3 on page 330. If you have questions, please bring them to class. Wednesday, November 12: #6 in section 46. Here are comments on test 5. Monday, November 10: #3 in section 34. Friday, November 7: ## 2(a),3 in section 33. Here is a copy of test 5. Wednesday, November 5: ## 1,2,6 from section 32. All problems from section 31 are very good. I advise to try as many as you have time for. Friday, October 31: ## 2,4,5(a),6 from section 30. Wednesday, October 29: ## 1,3,4,5,6 (you can actually see some of these problems show up on qual exams). Also, read the definitions and statements from section 29 on local compactness. This is the proof of Tychonoff's Theorem. Please read on your own, the proof is in the first 2 and 1/2 pages. And here is a comment on test 4. Monday, October 27: #2, page 181. Friday, October 24: ## 5,7 on page 171. Here is a copy of test 4. Wednesday, October 22: ## 2,3,4 on page 171. Monday, October 20: what are the connected components and path-components in each of the nine topologies on the three points set? (This is a good problem to test you staying honestly close to the definitions and not letting your intuition run wild). Also do #8 on page 163. Friday, October 17: 1,2,3 on pages 157-158. Here are the midterm problems and midterm solutions. And here are comments on test 3. Monday, October 12: ## 1,2 on page 152, then apply #2 to prove the following statement. Show that a ray [a, \infty) in the real line is connected. Answers to problem 4 and some comments that I don't want to spend class time on. Two problems related to the Wednesday class: ## 1,4 on pages 133, 134. Also, as I suggested, read the examples on pages 132-133. Here is a copy of test 3 (updated on Monday, October 13; also I remind you taht it's OK to turn in your test on Wednesday before the midterm). It is due in class on Monday, October 13. The midterm is on Wednesday, October 15. Monday, October 6: ## 4,5 on page 127. #4 is the long one, there are 9 answers to give in part (a) and 12 in part (b). Friday, October 3: ## 2,3 on page 126. Here are comments on test 2. And I am using this chance to post comments on test 1 that I forgot to do. Wednesday, October 1: you can see pictures I was drawing in class in section 22, with more details, Also do the great problem #1 on page 144. Here is the optional homework problem: suppose we are given two continuous maps f: X \to Z and g: Y \to Z, both in the category of topological spaces Top. Prove that the following topological space L is the limit of this diagram. L is the subset of X x Y with the subspace topology. It is defined as the set of all points (x,y) such that f(x) = g(y). It is called the "pullbac"k. Monday, September 29: ## 1,2,3,4,5,6 on page 118. Monday, September 22: ## 2,3,5,6,8 on page 111. Read Theorem 18.3 (the pasting lemma) before you do #8. It is an easy but very useful generalization of part (f) from the theorem in class. Here is a copy of test 2. It is due in class on Monday, September 29. Friday, September 19: ##10,11,12,13 on page 101 and #2 on page 111 Also solve the following problem: Suppose A is a closed subset of a topological space X. Show that the collection of open sets disjoint from A together with the open sets containing A form a topology on X (possibly coarser than the original topology). Let X/A stand for the set consisting of points of outside of A and one point for the whole equivalence class of points from A. Show that the topology in the first question allows you to define a topology on X/A. Prove that if X is a regular space then the topology on X/A is Hausdorff. Wednesday, September 17: ## 7,8,9 in section 17 Monday, September 15: ## 3,6 in section 17 Friday, September 12: ## 4,8,9,10 in section 16 Wednesday, September 10: ## 1,2,3 on pages 91-92. Also do these two problems: 1) Read the definition on page 86, then prove that B is a basis, without reading the next paragraph. 2) If B is a basis for the topology on X, for any subset Y of X we get the collection of all intersections of elements of B with Y. Show that this collection is a basis for the subspace topology on Y. Here is a copy of test 1. It is due in class on Monday, September 15. Monday, September 8: #5 (2nd statement). Friday, September 5: #5 (only the first sentence) and #8. Solution of problem 4(b). Wednesday, September 3: do #5 on page 83, also do #7. Here you can ignore the second topology T_2 or read about it at the top of page 82. Also, here is the problem from class: there are three topologies in the plane defined using three templates for the basis: open disks, open straight squares, and open diagonal squares. Why are these topologies the same topology? Homework after Friday, August 29: #4 from section 13 (still not about bases) and the following three problems I stated in class 1) Classify all topologies on the set with three elements up to homeomorphisms. This means you need to find out which pairs of topologies have pairs of continuous maps between them that are inverses of each other. (You might find it useful to use the easy fact that being homeomorphic is an equivalence relation. So what you are actually doing is sorting out the equivalence classes of these topologies.) 2) Suppose X is a topological space and U is a subset of X. Consider the collection of all intersections of U with the elements of the topology. Notice that all of these are subsets of U. Prove that this new collection of subsets of U is a topology (on U). 3) Prove that a topology (on some set X) is a basis for a topology (on X). Homework after Wednesday: #1,2,3 in section 13. Here is the link I promised to the archive of past prelims. When I use some of these topology problems, my references will be to dates in the archive.