10-327/Homework Assignment 4
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Reading
Read sections 23 through 25 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 26 through 27, just to get a feel for the future.
Doing
Solve and submit problems 1-3 and 8-10 Munkres' book, pages 157-158.
Due date
This assignment is due at the end of class on Monday, October 25, 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 world, given limited resources".
| Dror's notes above / Student's notes below |
Questions
1)Hi, quick question. I am wondering if the term test will cover the material on this assignment, or only the material before the assignment. Thanks! Jason.
2) In EXAMPLE 7 on page 151 Munkres claims that Rn~ is ['clearly' :)] homeomorphic to Rn: where Rn~ consists of all sequences x=(x1,x2,x3,...) with xi=0 for i>n, and Rn consists of all sequences x=(x1,x2,...xn). Why are they homeomorphic ?? Thank you kindly. Oliviu.
RE: 2) Let [math]\displaystyle{ F :\tilde R^n \rightarrow R^n }[/math] be defined as [math]\displaystyle{ F(x)= \prod_{i=1}^{n} \pi_i (x) }[/math] and let [math]\displaystyle{ F^{-1} : R^n \rightarrow \tilde R^n }[/math] be defined as [math]\displaystyle{ F^{-1}(x)= \prod_{i \in Z_+} f_i (x) }[/math] where [math]\displaystyle{ f_i (x) = \pi_i (x) }[/math] if [math]\displaystyle{ 1 \le i \le n }[/math] and [math]\displaystyle{ f_i(x)=0 }[/math] otherwise. Then both [math]\displaystyle{ F }[/math] and [math]\displaystyle{ F^{-1} }[/math] are continuous, because we are working in the product topology and the component functions, namely the projection function and constant function are continuous. Also [math]\displaystyle{ F }[/math] is a bijection because [math]\displaystyle{ F(F^{-1}(x_1, \ldots, x_n))=(x_1, \ldots, x_n) }[/math] and [math]\displaystyle{ F^{-1}(F(x_1, \ldots, x_n, 0,0, \ldots))=(x_1, \ldots, x_n, 0,0, \ldots) }[/math], i.e [math]\displaystyle{ F }[/math] has a left and right inverse. So [math]\displaystyle{ F }[/math] is a homeomorphism between the two spaces. Quick question is there a nicer way of writing math than using the math tag? Ian 16:03, 22 October 2010 (EDT)
3)Question. Suppose we have a function f going from topological space X to Y which is not onto and a function g going from Y to Z. Could I still define the composition of f and g? i.e. g circle f? -Kai Xwbdsb 19:19, 22 October 2010 (EDT)
- If I understand your question, I don't see why not...think about [math]\displaystyle{ \mathbb{R} }[/math] for example. [math]\displaystyle{ f(x)=x^2 }[/math] is not onto, then let [math]\displaystyle{ g(x)=e^x }[/math] then g compose f is [math]\displaystyle{ e^{x^2} }[/math] - John
- I agree but look at munkre's page 17 last sentence. Note that g compose with f is defined only when the range of f equals the domain of g. So I just want to confirm with Dror if there is something wrong here.
- Touche, I see your point...that is strange - John
- I agree but look at munkre's page 17 last sentence. Note that g compose with f is defined only when the range of f equals the domain of g. So I just want to confirm with Dror if there is something wrong here.
4)Question about the proof for [0,1] being connected. A few details are omitted. why would a closed subset of [0,1] contain its supremum? Also why [0,g_0] being a subset of A follows automatically after we showed that g_0 is in A? -Kai
- 1. Suppose [math]\displaystyle{ S }[/math] is closed in [math]\displaystyle{ [0,1]. \Rightarrow S^C }[/math] is open. If [math]\displaystyle{ sup(S)=\alpha \notin S \Rightarrow \exists r\gt 0 }[/math] s.t. [math]\displaystyle{ B(\alpha, r) \subset S^C \Rightarrow \alpha - 0.5r \in S^C \Rightarrow \alpha - 0.5r \lt \alpha }[/math] is an upper bound for S. [math]\displaystyle{ \Rightarrow \Leftarrow }[/math] Frank
