10-327/Homework Assignment 3: Difference between revisions
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*Question about HW3 8(b). I still don't understand why the uniform topology on <math>{\mathbb R}^\infty</math> is strictly finer than the product topology. If you find any open nbd in uniform topology of any point in <math>{\mathbb R}^\infty</math> only finitely many component are in the form of <math>(x-\epsilon,x+\epsilon)</math> because the sequence has infinitely many <math>0</math>'s. Can't I just choose these <math>(x-\epsilon,x+\epsilon)</math> multiply by infinitely many copies of <math>{\mathbb R}</math> in the product topology? -Kai |
*Question about HW3 8(b). I still don't understand why the uniform topology on <math>{\mathbb R}^\infty</math> is strictly finer than the product topology. If you find any open nbd in uniform topology of any point in <math>{\mathbb R}^\infty</math> only finitely many component are in the form of <math>(x-\epsilon,x+\epsilon)</math> because the sequence has infinitely many <math>0</math>'s. Can't I just choose these <math>(x-\epsilon,x+\epsilon)</math> multiply by infinitely many copies of <math>{\mathbb R}</math> in the product topology? -Kai |
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** Good thought, but there is something wrong in your logic. This though remains your assignment to do, so what I'll write may sound a bit cryptic: Note that in the uniform topology, the <math>(\pm\epsilon)</math> constraint applies also to the <math>0</math>'s. |
** Good thought, but there is something wrong in your logic. This though remains your assignment to do, so what I'll write may sound a bit cryptic: Note that in the uniform topology, the <math>(\pm\epsilon)</math> constraint applies also to the <math>0</math>'s. [[User:Drorbn|Drorbn]] 18:13, 12 October 2010 (EDT) |
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Revision as of 17:13, 12 October 2010
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Reading
Read sections 19, 20, 21, and 23 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 24 and 26, 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 on page 118, and problems 3, 4, 5, 6, 8, 9, 10 on pages 126-128.
Class Photo
Identify yourself in the 10-327/Class Photo page!
Due date
This assignment is due at the end of class on Thursday, October 14, 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 |
Discussion
- Question about HW3 8(b). I still don't understand why the uniform topology on [math]\displaystyle{ {\mathbb R}^\infty }[/math] is strictly finer than the product topology. If you find any open nbd in uniform topology of any point in [math]\displaystyle{ {\mathbb R}^\infty }[/math] only finitely many component are in the form of [math]\displaystyle{ (x-\epsilon,x+\epsilon) }[/math] because the sequence has infinitely many [math]\displaystyle{ 0 }[/math]'s. Can't I just choose these [math]\displaystyle{ (x-\epsilon,x+\epsilon) }[/math] multiply by infinitely many copies of [math]\displaystyle{ {\mathbb R} }[/math] in the product topology? -Kai
- Good thought, but there is something wrong in your logic. This though remains your assignment to do, so what I'll write may sound a bit cryptic: Note that in the uniform topology, the [math]\displaystyle{ (\pm\epsilon) }[/math] constraint applies also to the [math]\displaystyle{ 0 }[/math]'s. Drorbn 18:13, 12 October 2010 (EDT)
