Drorbn - User contributions [en]

10-327/Homework Assignment 4

2010-10-22T15:55:25Z

Oliviulazar:

{{10-327/Navigation}}

===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 {{10-327/vp|0927}}) in a manner similar to (say) {{dbnvp link|AKT-090910-1|AKT-090910-1}}, and/or add links to the blackboard shots, in a manner similar to {{dbnvp link|Alekseev-1006-1|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".

{{Template:10-327:Dror/Students Divider}}

===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.

10-327/Homework Assignment 3

2010-10-14T03:37:24Z

Oliviulazar:

{{10-327/Navigation}}

===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 {{10-327/vp|0927}}) in a manner similar to (say) {{dbnvp link|AKT-090910-1|AKT-090910-1}}, and/or add links to the blackboard shots, in a manner similar to {{dbnvp link|Alekseev-1006-1|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".

{{Template:10-327:Dror/Students Divider}}
===Discussion===

*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. [[User:Drorbn|Drorbn]] 18:13, 12 October 2010 (EDT)
***But once you get the intersection with <math>{\mathbb R}^\infty</math>, those constraints should disappear?
**** No. [[User:Drorbn|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. -Kai[[User:Xwbdsb|Xwbdsb]] 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 [[User:Xwbdsb|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

10-327/Homework Assignment 3

2010-10-14T03:36:27Z

Oliviulazar:

10-327/Homework Assignment 3

2010-10-14T03:35:44Z

Oliviulazar:

{{10-327/Navigation}}

===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 {{10-327/vp|0927}}) in a manner similar to (say) {{dbnvp link|AKT-090910-1|AKT-090910-1}}, and/or add links to the blackboard shots, in a manner similar to {{dbnvp link|Alekseev-1006-1|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".

{{Template:10-327:Dror/Students Divider}}
===Discussion===

*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. [[User:Drorbn|Drorbn]] 18:13, 12 October 2010 (EDT)
***But once you get the intersection with <math>{\mathbb R}^\infty</math>, those constraints should disappear?
**** No. [[User:Drorbn|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. -Kai[[User:Xwbdsb|Xwbdsb]] 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 [[User:Xwbdsb|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 :):)