# 10-327/Homework Assignment 2

### 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) , and/or add links to the blackboard shots, in a manner similar to . 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 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 ${\displaystyle x\times x}$ mean when ${\displaystyle x}$ is an element in ${\displaystyle X}$? Does the author mean the ordered pair ${\displaystyle (x,x)}$? And we assume that we put product topology on ${\displaystyle X\times X}$? -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. ${\displaystyle X\times X}$ is Hausdorff. We take any point in ${\displaystyle \Delta }$ complement. So we can separate it from any point in ${\displaystyle \Delta }$. But to separate it from the entire ${\displaystyle \Delta }$ 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)