Drorbn - User contributions [en]

10-327/Homework Assignment 5 Solutions

2010-12-20T03:51:25Z

Bcd:

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===Solution===
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An assignment without a solution is like a nightmare to me. I like every question accompanied
with a clean solution aside.(Might not be the case for research question because they are just
simply too hard.) I would like to share this happiness of understanding and acquiring knowledge
with everybody because I don't think this class is a battle. I certainly don't like the idea that
we should keep information/answers as something like business secrets. This learning process
should be enjoyable which should be full of discussions instead of things like "you have to think on
your own/ I can't tell you the answer". I know maybe other people might not agree with me but I believe
a positive learning environment is crucial to truly understanding something well although we should not neglect
independent thinking at the same time. That is why I share whatever I have with you. If I am wrong feel free to criticize
me, and I am pretty sure a lot of people don't agree with me. But that is OK because there is just no
absolute right or wrong and everybody is doing what they think is right. Just like you can't say if
Axiom of choice is right or not. If you believe then it is right. If you don't believe it then it is wrong.
Or maybe, for the entire subject--math, if you believe it then it is right. If you don't believe it then it
is completely wrong.-Kai

Too late for marks, but just in time for the holidays are the extra problems (1,6,8,12 pp 170-171) - just sketches:

1. (a) is trivial. For (b), suppose X is compact Hausdorff relative to topologies <math>T_1, T_2</math> which are comparable. The identity function Id is continuous in one direction and so takes compact sets to compact sets. The rest is easy: fix any open set U and consider Id(X-U) ...

6. Also trivial. As in 1(b), use (i) compact subsets of T2 spaces are closed and (ii) closed subsets of compact spaces are compact.

8. If f is continuous, fix any point (x,y) not on the graph. Separate y and f(x) by open sets U, V, respectively. By continuity, find W open in X st f(W) is a subset of V. Then W x U is open disjoint from the graph. Conversely, if the graph is closed, apply the hint ...

12. Fix an open cover <math>U_\alpha</math> for X. For each y in Y, its preimage under p is compact, so can be covered by some finite collection of the <math>U_\alpha</math>, which also gives an open set. Do this for every y in Y. But then apply the hint to notice that we get a covering of Y by open neighbourhoods; apply compactness and the hint to conclude we only care about the preimages of finitely many of these nbds in Y, which are already inside finitely many finite unions of open sets.

[[User:Bcd|Bcd]] 22:51, 19 December 2010 (EST)

10-327/Homework Assignment 8 Solutions

2010-12-19T19:40:03Z

Bcd:

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In order to make our topological space of homework assignments complete, I need to upload the solution to the last assignment.(Sorry for getting so busy and keep forgetting things) What metric we should put on it?...Its quite difficult to create a complete space...

[http://katlas.math.toronto.edu/drorbn/images/f/ff/10-327a801.JPG page1]
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-Kai [[User:Xwbdsb|Xwbdsb]] 07:32, 19 December 2010 (EST)

I think the set of homeworks is a finite point set, hence automatically compact ... if you view the homeworks as spaces instead of points, then the resulting product space is complete if each homework is complete, as we know ... exercise: (a) is the converse also true? (b) what about infinite products?

Anyway, here is another solution set, this one typed (though not graded, so use at your own risk): [[User:Bcd|Bcd]] 14:40, 19 December 2010 (EST)

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327-HW8.pdf]

File:10-327-HW8.pdf

2010-12-19T19:29:48Z

Bcd: typed solutions to HW 8

typed solutions to HW 8

10-327/Homework Assignment 7

2010-11-22T03:32:34Z

Bcd: /* Due date */

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===Reading===
'''Read''' sections <math>\{31,32,33\}</math> 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 <math>\emptyset</math>, just to get a feel for the future.

===Doing===
Solve and submit the following problems from Munkres' book:
* Problem 1 on page 199.
* Problem 1 on page 205.
* Problems 1, 4, 5, 8, 9 on pages 212-213.

'''Remark.''' The following fact, which we will prove later, may be used without a proof: If <math>X</math> is a topological space and <math>f_n:X\to[0,1]</math> are continuous functions, then the sum <math>f(x):=\sum_{n=1}^\infty\frac{f_n(x)}{2^n}</math> is convergent and defines a continuous function on <math>X</math>.

===Due date===
This assignment is due at the end of class on Thursday, November 25, 2010.

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*Question: In problem 1 p205, is asks us to show that any closed subspace of a normal space is also normal. Do we really need the condition that the subspace be closed? - Jdw
** Yes. [[User:Drorbn|Drorbn]] 19:14, 19 November 2010 (EST)

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Questions by Kai [[User:Xwbdsb|Xwbdsb]] 21:26, 19 November 2010 (EST) were moved to [[10-327/Classnotes for Thursday November 18|Classnotes for Thursday November 18]] as they are about that class and not about this assignment. [[User:Drorbn|Drorbn]] 06:03, 20 November 2010 (EST)

*Question. If we have a finite set of continuous function mapping from any topological space into the reals. Any linear combination of these continuous function is still continuous right? The proof is a little extension of 157 proof. This is used to prove the statement you mentioned above. -Kai[[User:Xwbdsb|Xwbdsb]] 17:14, 20 November 2010 (EST)
** Any linear combination of functions from <math>\mathcal{C}(X,\mathbb{R})</math> (an uncountable set unless X is empty) is continuous. On its own, however, this proves nothing about infinite sums. [[User:Bcd|Bcd]] 22:32, 21 November 2010 (EST)

*Question about 9. Is J any indexing set? Possibly uncountable? in the hint: A means any closed set? -Kai [[User:Xwbdsb|Xwbdsb]] 22:13, 20 November 2010 (EST)

10-327/Class Photo

2010-10-08T02:40:59Z

Bcd: /* Who We Are... */

Our class on September 30, 2010:

[[Image:10-327-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
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Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}



{{Photo Entry|last=Asher|first=Matt|userid=Asher|email=matt.asher utoronto.ca|location=[http://mattasher.com/images/Me_in_photo.jpg Under the arrow] |comments= }}

{{Photo Entry|last=Darwin|first=Ben|userid=bcd|email=[first.last] @ utoronto.ca|location=Back row, 2nd from left |comments= }}

{{Photo Entry|last=DiPoce|first=Christian|userid=Cdipoce|email=christian.dipoce@ utoronto.ca|location=very back row, 2nd from right |comments= black t-shirt, yellow writing; as handsome as ever.}}

{{Photo Entry|last=Dranovski|first=Anne|userid=Anne.d|email=a.dranovski@ utoronto.edu|location=front row, jean jacket. |comments= }}

{{Photo Entry|last=Iorgulescu|first=Ana|userid=Anai|email=ana.iorgulescu@ utoronto.ca|location=back row, 3rd from left|comments=}}

{{Photo Entry|last=Kang|first=Soo Min|userid=soomin_kang|email=soomin.kang@ utoronto.ca|location=2nd row (from front), 2nd (from left)|comments= eyes closed, beige windbreaker}}

{{Photo Entry|last=Milcak|first=Juraj|userid=milcak|email=j.milcak @ utoronto.edu|location=frontmost, rightmost.|comments= }}

{{Photo Entry|last=Woodfine|first=Jason|userid=Jdw|email=jason(dot)woodfine (at) utoronto(dot)ca|location=3rd Row (from front), 4th (from left).|comments= Grey collared shirt, arms crossed, glasses.}}

{{Photo Entry|last=Zhao|first=Frank|userid=Fzhao|email=frank.zhao@ utoronto.ca|location=2nd Row (from front), 6th (from left).|comments= The one in the blue shirt}}

{{Photo Entry|last=Yang|first=Kai|userid=xwbdsb|email=kai.b.yang@utoronto.ca|location=.|comments= Was planning to take the picture but was late because of washroom...}}
|}