10327/Homework Assignment 2

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 101102, and problems 7a, 7b, 8, 9ab, 9c, 13 on pages 111112.
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: Topology100927) in a manner similar to (say) AKT0909101, and/or add links to the blackboard shots, in a manner similar to Alekseev10061. 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 world, given limited resources".
Dror's notes above / Student's notes below 
Remark on the Due Date
 Dear Professor BarNatan, 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 mean when is an element in ? Does the author mean the ordered pair ? And we assume that we put product topology on ? 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. is Hausdorff. We take any point in complement. So we can separate it from any point in . But to separate it from the entire 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)
 I found a way to approach this problem but I am not sure about the technicality. is Hausdorff. We take any point in complement. So we can separate it from any point in . But to separate it from the entire 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)
 Yes and yes. Drorbn 07:19, 2 October 2010 (EDT)