Difference between revisions of "10-327/Homework Assignment 3"

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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
** 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)

Revision as of 18:13, 12 October 2010

Contents

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: dbnvp Topology-100927) in a manner similar to (say) dbnvp AKT-090910-1, and/or add links to the blackboard shots, in a manner similar to dbnvp 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 {\mathbb R}^\infty is strictly finer than the product topology. If you find any open nbd in uniform topology of any point in {\mathbb R}^\infty only finitely many component are in the form of (x-\epsilon,x+\epsilon) because the sequence has infinitely many 0's. Can't I just choose these (x-\epsilon,x+\epsilon) multiply by infinitely many copies of {\mathbb R} 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 (\pm\epsilon) constraint applies also to the 0's. Drorbn 18:13, 12 October 2010 (EDT)