10-327/Homework Assignment 3

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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 Monday, October 18, 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)
      • But once you get the intersection with {\mathbb R}^\infty, those constraints should disappear?
        • No. Drorbn 21:18, 12 October 2010 (EDT)
  • Hi Dror, 8(b)(c) are really difficult and they involve several different concepts about infinity. Do you think you can go through them in class? Different concepts of infinity seem to be really intricate and I don't know how to argue properly. It seems it is true either way evenly philosophical in my point of view. Although its my homework assignment to do I still want to learn how to argue in questions like that. I hope you can teach us for learning purposes rather than marks. Maybe some other smart guys probably have already figured that out but I am not as smart. But this doesn't change the fact that I want to learn. Thanks. -KaiXwbdsb 23:56, 12 October 2010 (EDT)
  • I really have been spending a whole day on this assignment but still nothing comes to my mind how to prove box topology is strictly finer than l^2 topology. Dror could you possibly explain on this for a little bit please?-Kai Xwbdsb 00:18, 13 October 2010 (EDT)
  • For a sequence to eventually end with 0's. does it mean that it can only have finitely many non zeros terms? How about for a sequence with infinitely many 0's does it mean that the sequence can only have finitely many non-0 terms? -Kai
    • I think the aswer to your first question is yes: Quote Munkres page 118 q7 " ... all sequences that are eventually zero, that is all sequences (x1, x2, ...) such that xi != 0 for finitely many values of i". The answer to the second question, I think is no: consider (1, 0, 1, 0, 1, 0 ...). Please correct me if I'm wrong. By the way, i've also been working on this homework for the past two days, and still did not finish... so I'm a dumb-dumb too :):) - Oliviu