10-327/Homework Assignment 4: Difference between revisions
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RE: 2) Let <math>F :\tilde R^n \rightarrow R^n</math> be defined as <math>F(x)= \prod_{i=1}^{n} \pi_i (x)</math> and let <math>F^{-1} : R^n \rightarrow \tilde R^n</math> be defined as <math>F^{-1}(x)= \prod_{i \in Z_+} f_i (x)</math> where <math> f_i (x) = \pi_i (x) </math> if <math> 1 \le i \le n </math> and <math> f_i(x)=0 </math> otherwise. Then both <math> F </math> and <math> 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> F </math> is a bijection because <math> F(F^{-1}(x_1, \ldots, x_n))=(x_1, \ldots, x_n) </math> and <math> F^{-1}(F(x_1, \ldots, x_n, 0,0, \ldots))=(x_1, \ldots, x_n, 0,0, \ldots) </math>, i.e <math> F </math> has a left and right inverse. So <math> F </math> is a homeomorphism between the two spaces. Quick question is there a nicer way of writing math than using the math tag? [[User:Ian|Ian]] 16:03, 22 October 2010 (EDT) |
RE: 2) Let <math>F :\tilde R^n \rightarrow R^n</math> be defined as <math>F(x)= \prod_{i=1}^{n} \pi_i (x)</math> and let <math>F^{-1} : R^n \rightarrow \tilde R^n</math> be defined as <math>F^{-1}(x)= \prod_{i \in Z_+} f_i (x)</math> where <math> f_i (x) = \pi_i (x) </math> if <math> 1 \le i \le n </math> and <math> f_i(x)=0 </math> otherwise. Then both <math> F </math> and <math> 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> F </math> is a bijection because <math> F(F^{-1}(x_1, \ldots, x_n))=(x_1, \ldots, x_n) </math> and <math> F^{-1}(F(x_1, \ldots, x_n, 0,0, \ldots))=(x_1, \ldots, x_n, 0,0, \ldots) </math>, i.e <math> F </math> has a left and right inverse. So <math> F </math> is a homeomorphism between the two spaces. Quick question is there a nicer way of writing math than using the math tag? [[User:Ian|Ian]] 16:03, 22 October 2010 (EDT) |
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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 [[User:Xwbdsb|Xwbdsb]] 19:19, 22 October 2010 (EDT) |
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Revision as of 18:19, 22 October 2010
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![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
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 be defined as and let be defined as where if and otherwise. Then both and are continuous, because we are working in the product topology and the component functions, namely the projection function and constant function are continuous. Also is a bijection because and , i.e has a left and right inverse. So 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)
