10-327/Homework Assignment 2

DBN: Classes: 2010-11: 10-327/Navigation Additions to the MAT 327 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 13 About This Class, Monday - Continuity and open sets, Thursday - topologies, continuity, bases. 2 Sep 20 Monday - More on bases, Thursdsay - Products, Subspaces, Closed sets, HW1, HW1 Solutions 3 Sep 27 Monday - the Cantor set, closures, Thursday, Class Photo, HW2, HW2 Solutions 4 Oct 4 Monday - the axiom of choice and infinite product spaces, Thursday - the box and the product topologies, metric spaces, HW3, HW3 Solutions 5 Oct 11 Monday is Thanksgiving. Thursday - metric spaces, sequencial closures, various products. Final exam's date announced on Friday. 6 Oct 18 Monday - connectedness in ${\mathbb {R} }$, HW4, HW4 Solutions, Thursday - connectedness, path-connectedness and products 7 Oct 25 Monday - Compactness of $[0,1]$, Term Test on Thursday, TT Solutions 8 Nov 1 Monday - compact is closed and bounded, maximal values, HW5, HW5 Solutions, Wednesday was the last date to drop this course, Thursday - compactness of products and in metric spaces, the FIP 9 Nov 8 Monday-Tuesday is Fall Break, Thursday - Tychonoff and a taste of Stone-Cech, HW6, HW6 Solutions 10 Nov 15 Monday - generalized limits, Thursday - Normal spaces and Urysohn's lemma, HW7, HW7 Solutions 11 Nov 22 Monday - $T_{3.5}$ and $I^{A}$, Thursday - Tietze's theorem 12 Nov 29 Monday - compactness in metric spaces, HW8, HW8 Solutions, Thursday - completeness and compactness 13 Dec 6 Monday - Baire spaces and no-where differentiable functions, Wednesday - Hilbert's 13th problem; also see December 2010 Schedule R Dec 13 See December 2010 Schedule F Dec 20 Final exam, Monday December 20, 2PM-5PM, at BR200 Register of Good Deeds Add your name / see who's in! See Hilbert's 13th

Reading

Read sections 17 through 21 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 22 through 24, just to get a feel for the future.

Doing

Solve the following problems from Munkres' book, though submit only the underlined ones: Problems 6, 7, 8, 13, 14, 19abc, 19d, 21 on pages 101-102, and problems 7a, 7b, 8, 9ab, 9c, 13 on pages 111-112.

Due date

This assignment is due at the end of class on Thursday, October 7, 2010.

Suggestions for Good Deeds

Annotate our Monday videos (starting with Video:  Topology-100927) in a manner similar to (say)  AKT-090910-1, and/or add links to the blackboard shots, in a manner similar to  Alekseev-1006-1. Also, make constructive suggestions to me, Dror and / or the videographer, Qian (Sindy) Li, on how to improve the videos and / or the software used to display them. Note that "constructive" means also, "something that can be implemented relatively easily in the real worlds, given limited resources".

Dror's notes above / Student's notes below

Remark on the Due Date

• Dear Professor Bar-Natan, October 5 seems like a Tuesday. Do you mean October 7, 2010? Thanks! Fzhao 23:42, 30 September 2010 (EDT)Frank
  • I stand corrected. Drorbn 06:33, 1 October 2010 (EDT)

Questions

• Hi, I have a quick question. In the last question on the assignment that is being marked, what does it mean for one function to "uniquely determine" another. Sorry, I have just never heard that terminology before. - Jdw
  • It means that any two functions with the property stated in the question are actually the same. Drorbn 07:19, 2 October 2010 (EDT)
• Xwbdsb 00:39, 2 October 2010 (EDT) I have a question about problem 13 on page 101. What does $x\times x$ mean when $x$ is an element in $X$? Does the author mean the ordered pair $(x,x)$? And we assume that we put product topology on $X\times X$? -Kai
  • Yes and yes. Drorbn 07:19, 2 October 2010 (EDT)
    • I found a way to approach this problem but I am not sure about the technicality. $X\times X$ is Hausdorff. We take any point in $\Delta$ complement. So we can separate it from any point in $\Delta$. But to separate it from the entire $\Delta$ we need to get the intersection of all its open nbds. Will that still be a valid open nbd? -Kai Xwbdsb 11:29, 2 October 2010 (EDT)
      • An arbitrary intersection of open sets is not necessarily open. This I'll say, but beyond this, it is your problem to solve. Drorbn 16:21, 2 October 2010 (EDT)