Drorbn - User contributions [en]

10-327/Classnotes for Monday December 6

2010-12-11T05:38:45Z

Anne.d:

{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101206-142909.jpg}}.

{{10-327/vp|1206}}

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

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:327-notes_for_120610.pdf Lecture Notes]

File:327-notes for 120610.pdf

2010-12-11T05:36:02Z

Anne.d: Notes on Baire spaces and no-where differentiable functions

Notes on Baire spaces and no-where differentiable functions

10-327/Homework Assignment 8 Solutions

2010-12-11T02:59:42Z

Anne.d:

10-327/Homework Assignment 7

2010-12-11T02:56:59Z

Anne.d: EDIT: moved Kai's HW7 solutions to new page

{{10-327/Navigation}}
===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.

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

*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)
** Yes, <math>J</math> is arbitrary and <math>A</math> is closed. [[User:Drorbn|Drorbn]] 06:41, 22 November 2010 (EST)

10-327/Homework Assignment 7 Solutions

2010-12-11T02:56:43Z

Anne.d:

{{10-327/Navigation}}

===Solution===
Here is the solution to HW7. -Kai
[http://katlas.math.toronto.edu/drorbn/images/b/b2/10-327a701_%281%29.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/3/30/10-327a701_%282%29.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/a/ac/10-327a701_%283%29.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a701_%284%29.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/4/40/10-327a701_%285%29.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/d/db/10-327a701_%286%29.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/e/ef/10-327a701_%287%29.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/6/66/10-327a701_%288%29.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/7/71/10-327a701_%289%29.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/6/62/10-327a701_%2810%29.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/9/97/10-327a701_%2811%29.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/0/0e/10-327a701_%2812%29.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/c/cc/10-327a701_%2813%29.JPG page13]
[http://katlas.math.toronto.edu/drorbn/images/1/1f/10-327a701_%2814%29.JPG page14]
[http://katlas.math.toronto.edu/drorbn/images/7/7c/10-327a701_%2815%29.JPG page15]
[http://katlas.math.toronto.edu/drorbn/images/1/13/10-327a701_%2816%29.JPG page16]
[http://katlas.math.toronto.edu/drorbn/images/1/14/10-327a701_%2817%29.JPG page17]
[http://katlas.math.toronto.edu/drorbn/images/8/88/10-327a701_%2818%29.JPG page18]

10-327/Homework Assignment 6 Solutions

2010-12-11T02:55:55Z

Anne.d:

{{10-327/Navigation}}

===Solution===
Here is the solution to HW6 -Kai
[http://katlas.math.toronto.edu/drorbn/images/2/2f/10-327a601.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/4/46/10-327a602.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/a/af/10-327a603.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/9/9f/10-327a604.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/f/fa/10-327a605.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/b/b1/10-327a606.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/1/1f/10-327a607.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/d/d2/10-327a608.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/f/fe/10-327a609.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/e/ea/10-327a610.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/5/54/10-327a611.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/a/ac/10-327a612.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/e/e1/10-327a613.JPG page13]

10-327/Homework Assignment 6

2010-12-11T02:55:41Z

Anne.d: EDIT: moved Kai's HW6 solutions to new page

{{10-327/Navigation}}
===Reading===
'''Read''' sections 37-38 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 30-33, just to get a feel for the future.

===Doing===
Solve and submit the following problems.

'''Problem 1.''' Problem 1 on page 235 of Munkres' book. (Though following the [[10-327/Errata_to_Munkres'_Book|Errata to Munkres' Book]], in part (c) replace <math>T_1</math> by <math>T_2</math>).

'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.

'''Problem 3.''' Show that any function <math>f\colon{\mathbb N}\to I^A</math> from the integers into a "cube" <math>I^A=[0,1]^A</math> has a unique continuous extension to <math>\beta{\mathbb N}</math>.

'''Problem 4.''' Use the fact that there is a countable dense subset within <math>I^I</math> to show that the cardinality of <math>\beta{\mathbb N}</math> is greater than or equal to the cardinality of <math>I^I</math>.

'''Problem 5.''' Show that the cardinality of <math>\beta{\mathbb N}</math> is also less than or equal to the cardinality of <math>I^I</math>, and therefore it is equal to the cardinality of <math>I^I</math>.

'''Problem 6.''' Show that if <math>\mu\in\beta{\mathbb N}\backslash{\mathbb N}</math> and if <math>\mbox{Lim}_\mu</math> is the corresponding generalized limit, and if <math>b</math> is a bounded sequence and <math>f\colon{\mathbb R}\to{\mathbb R}</math> is a continuous function, then <math>\mbox{Lim}_\mu f(b_k) = f(\mbox{Lim}_\mu b_k)</math>.

'''Problem 7.''' Show that there is no super-limit function <math>\mbox{SuperLim}</math> defined on bounded sequences of reals with values in the reals which has the following 4 properties:
# <math>\mbox{SuperLim}(a_k)=\lim a_k</math>, if the sequence <math>a_k</math> is convergent.
# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.
# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.
# <math>\mbox{SuperLim}(a'_k) = \mbox{SuperLim}(a_k)</math>, where <math>a'</math> is <math>a</math> "shifted once": <math>a'_k=a_{k+1}</math>.

===Due date===
This assignment is due at the end of class on Thursday, November 18, 2010.

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

Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that?
Why in problem 5 it says that the cardinality of a set is less or equal to another set so that the cardinality are the same?
[[User:Xwbdsb|Xwbdsb]] 00:25, 13 November 2010 (EST)
*One way to do this is that if you have a map <math>f(x):A \to B</math> such that f(x) is 1-1, then <math>A</math> has cardinality less than or equal to <math>B</math>. I haven't worked on this question yet, so it may be a red herring, but in general I find the technique useful. [[User:Johnfleming|John]]

Also I am wondering what is the super-limit because it wasn't done in class and it is not covered in the book too??[[User:Xwbdsb|Xwbdsb]] 00:36, 13 November 2010 (EST)
I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is
not the standard definition in our topology class? Sorry Dror I am sort of lost of what to do for this assignment...[[User:Xwbdsb|Xwbdsb]] 00:36, 13 November 2010 (EST)

Also what is the generalized limit? I search for this idea in the book but I didn't find anything. I think in the book before we understand Stone-Cech compatification theorem we need to understand what a regular space is and also we need to understand one-point compactification? Isn't Stone-Cech compatification just a special way to compatify the some topological space so that the continuous function with uniquely be extended to the compatification? [[User:Xwbdsb|Xwbdsb]] 00:58, 13 November 2010 (EST)

* I actually genuinely enjoyed writing this assignment, thinking that I've asked a lovely collection of questions that will challenge you in the positive sense of the word - make you scratch your heads, and struggle a bit, and fight a bit to understand what is going on and eventually get some satisfaction and gain some appreciation of the intricacy and beauty of the subject matter as you understand and solve these problems. So please challenge yourself and think and struggle a bit more. [[User:Drorbn|Drorbn]] 10:32, 13 November 2010 (EST)

* I have a question concerning Problem 7. Perhaps I am misunderstanding it, but if we just define <math>SuperLim(a_k)=0</math> for all sequences the 3 properties seem to hold, maybe that <math>SuperLim(a_k)=Lim(a_k)</math> if the sequence does converge should be added? Or maybe just ruling out constant functions is suffient... -[[User:Johnfleming|John]]
** Thanks! You are absolutely right. The question has been modified. [[User:Drorbn|Drorbn]] 15:23, 13 November 2010 (EST)

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Hi Dror, I understand you want to make it challenging but I think it is also a bit unfair for some students like me. I never learned super-limit function. It is not done in your lecture or discussed in the book. So solving this question totally relies on other background knowledge. I don't have such strong background compared to some other students and I didn't even have a serious lecture on cardinality. I am just hoping since you are asking these kind of questions could you possibly talk about it in class? Trust me Dror it is not that I am not working hard or not trying to struggle but my background is not even enough for me to understand the question. For example I don't know what super-limit function is. I couldn't find it in Munkres book I couldn't find it by googling? Are you talking about limsup?

And what is the generalized definition of limit? I cannot find the definition anywhere.[[User:Xwbdsb|Xwbdsb]] 10:01, 14 November 2010 (EST)

* In the question about super-limits you are asked to prove that "super-limits", objects with the given properties, do not exist. It is not surprising they are not in the literature. Cardinalities are covered in MAT246, which is a pre-requisite to this class, I believe. I'll say more about generalized limits in the coming class tomorrow. [[User:Drorbn|Drorbn]] 10:37, 14 November 2010 (EST)
** Munkres Chapter 1 is all about set theory. In particular sections 6,7 and 9 have discussions about cardinality, this may be a good resource if you are having difficulties with the ideas. - [[User:Johnfleming|John]]

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*A quick question (and I hope it isn't stupid...), but I'm not sure what "A" is when you talk about the cube <math>I^A</math>. What product of the unit interval are you talking about? --[[User:Wesley|Wesley]] 12:15, 14 November 2010 (EST)
** <math>A</math> is any set whatsoever. So the "cube" <math>I^A</math> is finite dimensional if <math>A</math> is finite, but it is infinite dimensional if <math>A</math> is infinite. [[User:Drorbn|Drorbn]] 13:28, 14 November 2010 (EST)

*Is it true that for question 6 the function f has to be bounded? Since even though b is a bounded sequence but f(b_k) might not be bounded so that the generalized limit is not defined. given that f is continuous. suppose f=1/x and b_k = 1/k. [[User:Xwbdsb|Xwbdsb]] 22:12, 15 November 2010 (EST)
** The function in the example you gave is not continuous; it not even defined at <math>x=0</math>. [[User:Drorbn|Drorbn]] 07:10, 16 November 2010 (EST)

* In problem 1 part c, the errata to Munkres says that should be Hausdorff.
** Thanks! [[User:Drorbn|Drorbn]] 07:10, 16 November 2010 (EST)

**Does the set of all bounded sequences has the same cardinality as the reals?[[User:Xwbdsb|Xwbdsb]] 15:00, 16 November 2010 (EST)

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Question About Problem 2

Should we assume that {0,1} has the discrete topology and that {0,1}^N has the corresponding product topology and should we also assume that C has the subspace topology that it inherits from R? Thank you.

* Yes and yes. [[User:Drorbn|Drorbn]] 06:59, 17 November 2010 (EST)

Question About Problem 3

If F if the extension of the function f to (beta)N, then do we need to ensure that if mu is in (beta)N \ N, then F(mu) is in [0,1]^A ? (This is similar to how in class we extended a bounded function b to b_tilda but then we did not guarantee that b_tilda was bounded so then the range for b_tilda can be R but may not be any closed interval in R.) Thank you.

<math>\tilde b </math> has to be bounded because <math>\beta{\mathbb N}</math> is compact. [[User:Ian|Ian]] 18:54, 17 November 2010 (EST)

Thank you Ian. What I meant to say was that if b: N -> [a,b] then we don't know for sure that b_tilda: beta_N -> [a,b] (ie the range of b_tilda could be [a-10, b+10] or something like this which is not contained in [a,b].) Thus, b_tilda is not necessarily an extension of b: N-> [a,b] but only an extension of b: N -> R. Do we need to take this into account for this question? Thanks.

Now I think that the range of b_tilda must be inside [a,b] since b_tilda(beta_N) = b_tilda(N_closure) which is contained in b_tilda(N)_closure = b(N)_closure which is contained in [a,b]_closure = [a,b]. Is this correct? Thanks.

* Yes. [[User:Drorbn|Drorbn]] 06:47, 18 November 2010 (EST)

10-327/Homework Assignment 5

2010-12-11T02:54:47Z

Anne.d: EDIT: moved Kai's HW5 solutions to new page

{{10-327/Navigation}}
===Reading===
'''Read''' sections 26 and 27 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 28 and 29, just to get a feel for the future.

===Doing===
Solve the following problems from Munkres' book, though submit only the underlined ones: Problems 1, 4, 5, 6, 7, 8, 9, 12 on pages 170-172, and problem 2 on page 177. (For the last, recall that <math>d(x,A):=\mbox{inf}_{a\in A}d(x,a)</math>).

===Due date===
This assignment is due at the end of class on Thursday, November 11, 2010.

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

*Question. In 4. By bounded metric space you mean there exists a point and an epsilon where this epsilon nbd contains everything in the metric space? -Kai
** Indeed so, though usually when talking about boundedness, people use the letter <math>M</math> and not the letter <math>\epsilon</math>. It makes no difference, of course.

10-327/Homework Assignment 5 Solutions

2010-12-11T02:54:36Z

Anne.d:

{{10-327/Navigation}}

===Solution===
[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a501.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a502.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/6/6b/10-327a503.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/6/61/10-327a504.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/1/1d/10-327a505.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/c/c6/10-327a506.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a507.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/d/df/10-327a508.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/9/9d/10-327a509.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/8/8e/10-327a510.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/7/75/10-327a511.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/7/70/10-327a512.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/d/dc/10-327a513.JPG page13]
[http://katlas.math.toronto.edu/drorbn/images/4/4d/10-327a514.JPG page14]
[http://katlas.math.toronto.edu/drorbn/images/c/c5/10-327a515.JPG page15]
[http://katlas.math.toronto.edu/drorbn/images/6/6d/10-327a516.JPG page16]
[http://katlas.math.toronto.edu/drorbn/images/c/c6/10-327a517.JPG page17]
[http://katlas.math.toronto.edu/drorbn/images/d/d5/10-327a518.JPG page18]

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

10-327/Navigation

2010-12-11T02:52:00Z

Anne.d:

<noinclude>Back to [[10-327]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 13
|[[10-327/About This Class|About This Class]], [[10-327/Classnotes for Monday September 13|Monday]] - Continuity and open sets, [[10-327/Classnotes for Thursday September 16|Thursday]] - topologies, continuity, bases.
|- align=left
|align=center|2
|Sep 20
|[[10-327/Classnotes for Monday September 20|Monday]] - More on bases, [[10-327/Classnotes for Thursday September 24|Thursdsay]] - Products, Subspaces, Closed sets, [[10-327/Homework Assignment 1|HW1]], [[10-327/Homework Assignment 1 Solutions|HW1 Solutions]]
|- align=left
|align=center|3
|Sep 27
|[[10-327/Classnotes for Monday September 27|Monday]] - the Cantor set, closures, [[10-327/Classnotes for Thursday September 30|Thursday]], [[10-327/Class Photo|Class Photo]], [[10-327/Homework Assignment 2|HW2]], [[10-327/Homework Assignment 2 Solutions|HW2 Solutions]]
|- align=left
|align=center|4
|Oct 4
|[[10-327/Classnotes for Monday October 4|Monday]] - the axiom of choice and infinite product spaces, [[10-327/Classnotes for Thursday October 7|Thursday]] - the box and the product topologies, metric spaces, [[10-327/Homework Assignment 3|HW3]], [[10-327/Homework Assignment 3 Solutions|HW3 Solutions]]
|- align=left
|align=center|5
|Oct 11
|Monday is Thanksgiving. [[10-327/Classnotes for Thursday October 14|Thursday]] - metric spaces, sequencial closures, various products. Final exam's date announced on Friday.
|- align=left
|align=center|6
|Oct 18
|[[10-327/Classnotes for Monday October 18|Monday]] - connectedness in <math>{\mathbb R}</math>, [[10-327/Homework Assignment 4|HW4]], [[10-327/Homework Assignment 4 Solutions|HW4 Solutions]], [[10-327/Classnotes for Thursday October 21|Thursday]] - connectedness, path-connectedness and products
|- align=left
|align=center|7
|Oct 25
|[[10-327/Classnotes for Monday October 25|Monday]] - compactness of <math>[0,1]</math>, [[10-327/Term Test|Term Test]] on Thursday, [[10-327/Term Test Solutions|TT Solutions]]
|- align=left
|align=center|8
|Nov 1
|[[10-327/Classnotes for Monday November 1|Monday]] - compact is closed and bounded, maximal values, [[10-327/Homework Assignment 5|HW5]], [[10-327/Homework Assignment 5 Solutions|HW5 Solutions]], Wednesday was the last date to drop this course, [[10-327/Classnotes for Thursday November 4|Thursday]] - compactness of products and in metric spaces, the FIP
|- align=left
|align=center|9
|Nov 8
|Monday-Tuesday is Fall Break, [[10-327/Classnotes for Thursday November 11|Thursday]] - Tychonoff and a taste of Stone-Cech, [[10-327/Homework Assignment 6|HW6]], [[10-327/Homework Assignment 6 Solutions|HW6 Solutions]]
|- align=left
|align=center|10
|Nov 15
|[[10-327/Classnotes for Monday November 15|Monday]] - generalized limits, [[10-327/Classnotes for Thursday November 18|Thursday]] - Normal spaces and Urysohn's lemma, [[10-327/Homework Assignment 7|HW7]], [[10-327/Homework Assignment 7 Solutions|HW7 Solutions]]
|- align=left
|align=center|11
|Nov 22
|[[10-327/Classnotes for Monday November 22|Monday]] - <math>T_{3.5}</math> and <math>I^A</math>, [[10-327/Classnotes for Thursday November 25|Thursday]] - Tietze's theorem
|- align=left
|align=center|12
|Nov 29
|[[10-327/Classnotes for Monday November 29|Monday]] - compactness in metric spaces, [[10-327/Homework Assignment 8|HW8]], [[10-327/Homework Assignment 8 Solutions|HW8 Solutions]], [[10-327/Classnotes for Thursday December 2|Thursday]] - completeness and compactness
|- align=left
|align=center|13
|Dec 6
|[[10-327/Classnotes for Monday December 6|Monday]] - Baire spaces and no-where differentiable functions, [[10-327/Classnotes for Wednesday December 8|Wednesday]] - Hilbert's 13th problem; also see [[December 2010 Schedule]]
|- align=left
|align=center|R
|Dec 13
|See [[December 2010 Schedule]]
|- align=left
|align=center|F
|Dec 20
|Final exam, Monday December 20, 2PM-5PM, at BR200
|- align=left
|colspan=3 align=center|[[10-327/Register of Good Deeds|Register of Good Deeds]] / [[10-327/To Do|To Do List]]
|- align=left
|colspan=3 align=center|[[Image:10-327-ClassPhoto.jpg|310px]] [[10-327/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-327-Splash.png|310px]] See {{Home Link|Talks/Fields-0911/|Hilbert's 13th}}
|}

10-327/Term Test

2010-12-11T02:48:53Z

Anne.d: EDIT: moved Kai's TT solutions to new page

{{10-327/Navigation}}

===Results===
The term test was held on Thursday October 28, 2010; it is available at {{Home link|classes/1011/327-Topology/TT.pdf|TT.pdf}}, and all marks are available on the annoying [https://portal.utoronto.ca/ UofT Portal].

The average grade was 72 and the standard deviation 26.5, but this does not tell the story right. The results are highly polarized; here is the full list of grades:

<blockquote>100 100 100 100 100 100 100 100 100 100 100 100 98 98 88 87 86 84 82 81 77 73 72 70 66 63 63 55 53 52 50 49 45 45 41 41 34 28 28 25 14</blockquote>

To me this means that there is a large group of students who are on top of things, and for whom that was an easy exam. Yet there is also a smaller group of students (say, those with grades near 50 or below) who either did not study or do not have the required background. These students should feel very alarmed. If you are one of them and you do not have a truly realistic plan to turn things around (it is not too late), you should quit this class by the faculty deadline of November 3rd, before it becomes an unnecessary dark spot on your transcript.

If your grade is 100, you have nothing to learn from this exam. (That may be a curse in disguise! Don't get over-confident!). If it is anything less, you missed something, even if something small. At any rate, you should read your exam carefully to see what that something is and to come up with a plan to fix it for the final, so that your grade then will be 100.

Note that problems with writing are problems, period. Perhaps you got a low grade but you feel you know the material enough for a high grade only you didn't write everything you know or you didn't it write well enough or the silly graders simply didn't get what you wrote (and it isn't a simple misunderstanding - see "appeals" below). If this describes you, don't underestimate your problem. If you don't process and resolve it, it is likely to recur.

====Appeals====
Remember! Grading is a difficult process and mistakes '''always''' happen - solutions get misread, parts are forgotten, grades are not added up correctly. You '''must''' read your exam and make sure that you understand how it was graded. If you disagree with anything, don't hesitate to complain! Your first stop should be the person who graded the problem in question, and only if you can't agree with him you should appeal to {{Dror}}.

Problems 3 and 4 were graded by {{Dror}}. All other problems were graded by David Reiss.

The deadline to start the appeal process is Thursday November 11 at 4PM.

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

10-327/Term Test Solutions

2010-12-11T02:48:33Z

Anne.d: EDIT: moved Kai's TT solutions to new page

{{10-327/Navigation}}

===Solutions===
[http://katlas.math.toronto.edu/drorbn/images/9/96/10-327midterm01.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/0/0a/10-327midterm02.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/e/eb/10-327midterm03.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/5/53/10-327midterm04.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/8/87/10-327midterm05.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/b/b3/10-327midterm06.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/9/93/10-327midterm07.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/6/6b/10-327midterm08.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/6/6f/10-327midterm09.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/2/22/10-327midterm10.JPG page10]
Here is the solution to Midterm exam 2010. Pretty straight forward and hope the final exam will not
be drastically opposite---extremely hard....-Kai

10-327/Navigation

2010-12-11T02:46:21Z

Anne.d:

<noinclude>Back to [[10-327]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 13
|[[10-327/About This Class|About This Class]], [[10-327/Classnotes for Monday September 13|Monday]] - Continuity and open sets, [[10-327/Classnotes for Thursday September 16|Thursday]] - topologies, continuity, bases.
|- align=left
|align=center|2
|Sep 20
|[[10-327/Classnotes for Monday September 20|Monday]] - More on bases, [[10-327/Classnotes for Thursday September 24|Thursdsay]] - Products, Subspaces, Closed sets, [[10-327/Homework Assignment 1|HW1]], [[10-327/Homework Assignment 1 Solutions|HW1 Solutions]]
|- align=left
|align=center|3
|Sep 27
|[[10-327/Classnotes for Monday September 27|Monday]] - the Cantor set, closures, [[10-327/Classnotes for Thursday September 30|Thursday]], [[10-327/Class Photo|Class Photo]], [[10-327/Homework Assignment 2|HW2]], [[10-327/Homework Assignment 2 Solutions|HW2 Solutions]]
|- align=left
|align=center|4
|Oct 4
|[[10-327/Classnotes for Monday October 4|Monday]] - the axiom of choice and infinite product spaces, [[10-327/Classnotes for Thursday October 7|Thursday]] - the box and the product topologies, metric spaces, [[10-327/Homework Assignment 3|HW3]], [[10-327/Homework Assignment 3 Solutions|HW3 Solutions]]
|- align=left
|align=center|5
|Oct 11
|Monday is Thanksgiving. [[10-327/Classnotes for Thursday October 14|Thursday]] - metric spaces, sequencial closures, various products. Final exam's date announced on Friday.
|- align=left
|align=center|6
|Oct 18
|[[10-327/Classnotes for Monday October 18|Monday]] - connectedness in <math>{\mathbb R}</math>, [[10-327/Homework Assignment 4|HW4]], [[10-327/Homework Assignment 4 Solutions|HW4 Solutions]], [[10-327/Classnotes for Thursday October 21|Thursday]] - connectedness, path-connectedness and products
|- align=left
|align=center|7
|Oct 25
|[[10-327/Classnotes for Monday October 25|Monday]] - compactness of <math>[0,1]</math>, [[10-327/Term Test|Term Test]] on Thursday, [[10-327/Term Test Solutions|TT Solutions]]
|- align=left
|align=center|8
|Nov 1
|[[10-327/Classnotes for Monday November 1|Monday]] - compact is closed and bounded, maximal values, [[10-327/Homework Assignment 5|HW5]], Wednesday was the last date to drop this course, [[10-327/Classnotes for Thursday November 4|Thursday]] - compactness of products and in metric spaces, the FIP
|- align=left
|align=center|9
|Nov 8
|Monday-Tuesday is Fall Break, [[10-327/Classnotes for Thursday November 11|Thursday]] - Tychonoff and a taste of Stone-Cech, [[10-327/Homework Assignment 6|HW6]]
|- align=left
|align=center|10
|Nov 15
|[[10-327/Classnotes for Monday November 15|Monday]] - generalized limits, [[10-327/Classnotes for Thursday November 18|Thursday]] - Normal spaces and Urysohn's lemma, [[10-327/Homework Assignment 7|HW7]]
|- align=left
|align=center|11
|Nov 22
|[[10-327/Classnotes for Monday November 22|Monday]] - <math>T_{3.5}</math> and <math>I^A</math>, [[10-327/Classnotes for Thursday November 25|Thursday]] - Tietze's theorem
|- align=left
|align=center|12
|Nov 29
|[[10-327/Classnotes for Monday November 29|Monday]] - compactness in metric spaces, [[10-327/Homework Assignment 8|HW8]], [[10-327/Classnotes for Thursday December 2|Thursday]] - completeness and compactness
|- align=left
|align=center|13
|Dec 6
|[[10-327/Classnotes for Monday December 6|Monday]] - Baire spaces and no-where differentiable functions, [[10-327/Classnotes for Wednesday December 8|Wednesday]] - Hilbert's 13th problem; also see [[December 2010 Schedule]]
|- align=left
|align=center|R
|Dec 13
|See [[December 2010 Schedule]]
|- align=left
|align=center|F
|Dec 20
|Final exam, Monday December 20, 2PM-5PM, at BR200
|- align=left
|colspan=3 align=center|[[10-327/Register of Good Deeds|Register of Good Deeds]] / [[10-327/To Do|To Do List]]
|- align=left
|colspan=3 align=center|[[Image:10-327-ClassPhoto.jpg|310px]] [[10-327/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-327-Splash.png|310px]] See {{Home Link|Talks/Fields-0911/|Hilbert's 13th}}
|}

10-327/Homework Assignment 4 Solutions

2010-12-11T02:44:35Z

Anne.d: /* Solutions */

{{10-327/Navigation}}

===Solutions===

[http://katlas.math.toronto.edu/drorbn/images/e/ec/10-327a401.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/5/59/10-327a402.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/1/18/10-327a403.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/a/a0/10-327a404.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/b/bb/10-327a405.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/e/e6/10-327a406.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a407.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/f/ff/10-327a408.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/1/1c/10-327a409.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/d/d8/10-327a410.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/8/82/10-327a411.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/d/db/10-327a412.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/5/56/10-327a413.JPG page13]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a414.JPG page14]

Math is like science. It is precise to the maximum. So whenever I write something in math I always want to make sure
that I am clear enough. Sets are sets functions are functions numbers are numbers. Math is also like art. It is a series
of arguments that we present to people to convince them what we believe is true. To some extent it is like politics or like
philosophy. I like to make my arguments simple, supplemented with diagrams and illustrations so that whenever I read
it I will be happy to believe what is written is true. However I still sin sometimes quoting big theorems without
actually fully understand them. But I try my best not to and present the simplest arguments that anybody could understand.-Kai

10-327/Homework Assignment 4 Solutions

2010-12-11T02:44:17Z

Anne.d: /* Solutions */

{{10-327/Navigation}}

===Solutions===

But the solution still goes here!
[http://katlas.math.toronto.edu/drorbn/images/e/ec/10-327a401.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/5/59/10-327a402.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/1/18/10-327a403.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/a/a0/10-327a404.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/b/bb/10-327a405.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/e/e6/10-327a406.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a407.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/f/ff/10-327a408.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/1/1c/10-327a409.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/d/d8/10-327a410.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/8/82/10-327a411.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/d/db/10-327a412.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/5/56/10-327a413.JPG page13]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a414.JPG page14]

Math is like science. It is precise to the maximum. So whenever I write something in math I always want to make sure
that I am clear enough. Sets are sets functions are functions numbers are numbers. Math is also like art. It is a series
of arguments that we present to people to convince them what we believe is true. To some extent it is like politics or like
philosophy. I like to make my arguments simple, supplemented with diagrams and illustrations so that whenever I read
it I will be happy to believe what is written is true. However I still sin sometimes quoting big theorems without
actually fully understand them. But I try my best not to and present the simplest arguments that anybody could understand.-Kai

10-327/Homework Assignment 4

2010-12-11T02:42:20Z

Anne.d: /* Solutions */

{{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===
# 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.
#* The term test will cover everything including Monday October 25 and including this assignment. [[User:Drorbn|Drorbn]] 13:42, 23 October 2010 (EDT)
# 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.

RE: 2) Let <math>F :\tilde R^n \rightarrow R^n</math> be defined as <math>F(x)= \prod_{i=1}^{n} \pi_i (x)</math> and let <math>F^{-1} : R^n \rightarrow \tilde R^n</math> be defined as <math>F^{-1}(x)= \prod_{i \in Z_+} f_i (x)</math> where <math> f_i (x) = \pi_i (x) </math> if <math> 1 \le i \le n </math> and <math> f_i(x)=0 </math> otherwise. Then both <math> F </math> and <math> F^{-1} </math> are continuous, because we are working in the product topology and the component functions, namely the projection function and constant function are continuous. Also <math> F </math> is a bijection because <math> F(F^{-1}(x_1, \ldots, x_n))=(x_1, \ldots, x_n) </math> and <math> F^{-1}(F(x_1, \ldots, x_n, 0,0, \ldots))=(x_1, \ldots, x_n, 0,0, \ldots) </math>, i.e <math> F </math> has a left and right inverse. So <math> F </math> is a homeomorphism between the two spaces. Quick question is there a nicer way of writing math than using the math tag? [[User:Ian|Ian]] 16:03, 22 October 2010 (EDT)
**Hi Ian I don't believe what you have said.</math>\tilde R^n</math> has the subspace topology from </math>R^\omega</math> where we indeed put product topology on it. This is not the same as </math>\tilde R^n</math> has the product topology. -Kai
***Kai I don't understand what you mean by not the same can you clarify? Do you mean they are not homeomorphic, i.e. there is something wrong with the functions I provided? [[User:Ian|Ian]] 17:47, 27 October 2010 (EDT)
****Hi Kai I see what you are trying to say now. Theorem 19.3 suggests they are they same. [[User:Ian|Ian]] 19:49, 27 October 2010 (EDT)

3)Question. Suppose we have a function f going from topological space X to Y which is not onto and a function g going from Y to Z. Could I still define the composition of f and g? i.e. g circle f? -Kai [[User:Xwbdsb|Xwbdsb]] 19:19, 22 October 2010 (EDT)
*If I understand your question, I don't see why not...think about <math>\mathbb{R}</math> for example. <math>f(x)=x^2</math> is not onto, then let <math>g(x)=e^x</math> then g compose f is <math>e^{x^2}</math> - John
**I agree but look at munkre's page 17 last sentence. Note that g compose with f is defined only when the range of f equals the domain of g. So I just want to confirm with Dror if there is something wrong here.
***Touche, I see your point...that is strange - John

4)Question about the proof for [0,1] being connected. A few details are omitted. why would a closed subset of [0,1] contain its supremum? Also why [0,g_0] being a subset of A follows automatically after we showed that g_0 is in A? -Kai
*1. Suppose <math>S </math> is closed in <math> [0,1]. \Rightarrow S^C </math> is open. If <math> sup(S)=\alpha \notin S \Rightarrow \exists r>0 </math> s.t. <math> B(\alpha, r) \subset S^C \Rightarrow \alpha - 0.5r \in S^C \Rightarrow \alpha - 0.5r < \alpha </math> is an upper bound for S. <math> \Rightarrow \Leftarrow </math>
*2. Recall that <math> G = \{g | [0,g] \subset A\}; g_0 = sup(G) \Rightarrow \forall g < g_0, [0,g] \subset A \Rightarrow [0, g_0) \subset A </math>. So, if <math> g_0 \in A \Rightarrow [0,g_0] \subset A</math>. -Frank [[User:Fzhao|Fzhao]] 23:50, 22 October 2010 (EDT)
**Thanks Frank. But I don't think your solution is convincing enough. \alpha - 0.5r is indeed not in S but why can you say it is an upper bound for S? Remember S could be rather complicated set all you know is that it is closed.
for 2 why is \Rightarrow \forall g < g_0, [0,g]? even if g_0 is sup(G) that does not mean anything less than g_0 would be in G. Consider [0,1] union {3}.

*Well, for the first question, not only is <math> \alpha - 0.5r </math> not in S, but neither is anything in <math> B(\alpha,r) </math>, since <math> S^C </math> is open. There can be no elements <math> \geq \alpha </math> in S because it's the supremum. Recall also we're working in the Reals. <math> \Rightarrow [\alpha - 0.5r, \infty) \subset S^C. </math>

*For the second question, notice that the supremum is the least upper bound (of G), so <math> \forall r > 0, \exists g \in G \cap B(g_0,r) \Rightarrow [0,g] \subset A \Rightarrow [0,g'] \subset A, \forall g' < g \Rightarrow g' \in G \Rightarrow \forall g < g_0 </math>, take <math> r = 0.5(g_0 - g) \Rightarrow \exists a \in (g_0 - r, g_0) \cap G, \Rightarrow [0,g] \subset [0,a] \subset A.</math>
Specifically, in your counterexample, if 3 is in G, then anything less than 3 is also in G by construction of G.

*Perhaps (I'm guessing here) you might have found supremum to be a confusing notion. If this is the case, have no fear, there's a chapter on supremum in Spivak's book Calculus. You can probably find one in the math library. - Frank [[User:Fzhao|Fzhao]] 09:58, 23 October 2010 (EDT)

*Question 1(b) on HW4. x stands for a point in S^1 but what does it mean by -x? -Kai
** <math>S^1\subset{\mathbb R}^2</math>, and <math>{\mathbb R}^2</math> is a vector space. [[User:Drorbn|Drorbn]] 13:42, 23 October 2010 (EDT)

*Question. During the proof of A \subset B \subset of A closure when they are all subspaces of X. If A is connected and so is B. I think there is loss of generality. We should prove any non-empty clopen set is B. But we are only proving those clopen sets whose intersection with A is non-empty are B. That is not enough because we could also have non-empty clopen set in B which does not intersect A. -Kai
** If <math>B</math> doesn't intersect <math>A</math> it's complement will. [[User:Drorbn|Drorbn]] 13:42, 23 October 2010 (EDT)

*Question. Consider the usual basis for infinite product topology that we put on an infinite cartesian product of topological spaces. The basis is just the topology generated minus the empty set right? [[User:Xwbdsb|Xwbdsb]] 23:23, 23 October 2010 (EDT)
** No. In an infinite (or even finite) product there are open sets which are unions of cylinders but are not themselves cylinders. A ball in the plane is a union of boxes but it is not a box. Another example is <math>{\mathbb R}^\omega-\{\bar{0}\}</math>. [[User:Drorbn|Drorbn]] 11:45, 24 October 2010 (EDT)

* Another question. We proved before if we have a collection of metrizable topological spaces then if we put cylinder topology on the infinite cartesian product then the resulting topological space is going to be metrizable. Then why isnt R^R with cylinder topology metrizable a contradiction since each copy of R is metrizable and R^R is just the infinite cartesian product of copies of R with indexing set R. -Kai[[User:Xwbdsb|Xwbdsb]] 22:58, 24 October 2010 (EDT)
** The theorem in class was about ''countable'' products of metric spaces. [[User:Drorbn|Drorbn]] 10:00, 25 October 2010 (EDT)
***For the first half of the proof we don't require countability assumption right because if the product space is metrizable even if it is uncountable product we can still create a homeomorphic subspace to each of the individual topological spaces? -Kai [[User:Xwbdsb|Xwbdsb]] 10:37, 25 October 2010 (EDT)

* Is it true that if one general topological space is metrizable and if one general topological space is homeomorphic to it then it is metrizable? Why? Metrizability is not just topological property? It involves the definition of metric which depends on the structure of a set.?? -Kai [[User:Xwbdsb|Xwbdsb]] 00:25, 25 October 2010 (EDT)
** "Metrizable" is a topological property, it means "having some metric which induces the topology". If <math>X</math> is metrizable and <math>Y</math> is homeomorphic to it, a metric on <math>X</math> composed with the homeomorphism from <math>Y</math> to <math>X</math> defines a metric on <math>Y</math> which induces the topology of <math>Y</math>. [[User:Drorbn|Drorbn]] 10:00, 25 October 2010 (EDT)

*Question. When proving if X is a metrizable space and A is a countably infinite set in X. The cardinality of the sequential closure of A is less or equal to the cardinality of R we used the fact that a metrizable space is Hausdorff right? Or you can say that the number of limit points should be less or equal to the number of converging sequences... -Kai
** Yes we use the fact that metric spaces are <math>T_2</math>. [[User:Drorbn|Drorbn]] 20:30, 26 October 2010 (EDT)

*Question. When proving topologist's sine curve is not path connected in page 157. Are we using the fact that for a cts function in Euclidean space if x_n converges to x then f(x_n) has to converge to f(x)? i.e. any open nbd of f(x) contains all but finitely many terms of f(x_n)? -Kai[[User:Xwbdsb|Xwbdsb]] 14:01, 27 October 2010 (EDT)
** We did not complete the proof of that fact in class. There are many ways to complete it, some use what you wrote and some don't. [[User:Drorbn|Drorbn]] 14:48, 27 October 2010 (EDT)
*Question. Is f(t) = (t,t,t...) cts in the R^\omega with uniform topology?.... -Kai [[User:Xwbdsb|Xwbdsb]] 15:23, 27 October 2010 (EDT)
** Yes. [[User:Drorbn|Drorbn]] 06:58, 28 October 2010 (EDT)

===Solutions===

Anne maybe you can help me again to create a page of solution. I tried but I don't know how to do so.
*Kai: page created, content moved. Just edit the navigation to create a new page. [[User:Anne.d|Anne.d]] 21:42, 10 December 2010 (EDT)

10-327/Homework Assignment 4 Solutions

2010-12-11T02:36:22Z

Anne.d:

{{10-327/Navigation}}

===Solutions===

Anne maybe you can help me again to create a page of solution. I tried but I don't know how to do so.
But the solution still goes here!
[http://katlas.math.toronto.edu/drorbn/images/e/ec/10-327a401.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/5/59/10-327a402.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/1/18/10-327a403.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/a/a0/10-327a404.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/b/bb/10-327a405.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/e/e6/10-327a406.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a407.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/f/ff/10-327a408.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/1/1c/10-327a409.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/d/d8/10-327a410.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/8/82/10-327a411.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/d/db/10-327a412.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/5/56/10-327a413.JPG page13]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a414.JPG page14]

Math is like science. It is precise to the maximum. So whenever I write something in math I always want to make sure
that I am clear enough. Sets are sets functions are functions numbers are numbers. Math is also like art. It is a series
of arguments that we present to people to convince them what we believe is true. To some extent it is like politics or like
philosophy. I like to make my arguments simple, supplemented with diagrams and illustrations so that whenever I read
it I will be happy to believe what is written is true. However I still sin sometimes quoting big theorems without
actually fully understand them. But I try my best not to and present the simplest arguments that anybody could understand.-Kai

10-327/Homework Assignment 4 Solutions

2010-12-11T02:35:34Z

Anne.d:

===Solutions===

Anne maybe you can help me again to create a page of solution. I tried but I don't know how to do so.
But the solution still goes here!
[http://katlas.math.toronto.edu/drorbn/images/e/ec/10-327a401.JPG page1]
[http://katlas.math.toronto.edu/drorbn/images/5/59/10-327a402.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/1/18/10-327a403.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/a/a0/10-327a404.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/b/bb/10-327a405.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/e/e6/10-327a406.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a407.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/f/ff/10-327a408.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/1/1c/10-327a409.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/d/d8/10-327a410.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/8/82/10-327a411.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/d/db/10-327a412.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/5/56/10-327a413.JPG page13]
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a414.JPG page14]

Math is like science. It is precise to the maximum. So whenever I write something in math I always want to make sure
that I am clear enough. Sets are sets functions are functions numbers are numbers. Math is also like art. It is a series
of arguments that we present to people to convince them what we believe is true. To some extent it is like politics or like
philosophy. I like to make my arguments simple, supplemented with diagrams and illustrations so that whenever I read
it I will be happy to believe what is written is true. However I still sin sometimes quoting big theorems without
actually fully understand them. But I try my best not to and present the simplest arguments that anybody could understand.-Kai

10-327/Navigation

2010-12-11T02:33:04Z

Anne.d:

<noinclude>Back to [[10-327]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 13
|[[10-327/About This Class|About This Class]], [[10-327/Classnotes for Monday September 13|Monday]] - Continuity and open sets, [[10-327/Classnotes for Thursday September 16|Thursday]] - topologies, continuity, bases.
|- align=left
|align=center|2
|Sep 20
|[[10-327/Classnotes for Monday September 20|Monday]] - More on bases, [[10-327/Classnotes for Thursday September 24|Thursdsay]] - Products, Subspaces, Closed sets, [[10-327/Homework Assignment 1|HW1]], [[10-327/Homework Assignment 1 Solutions|HW1 Solutions]]
|- align=left
|align=center|3
|Sep 27
|[[10-327/Classnotes for Monday September 27|Monday]] - the Cantor set, closures, [[10-327/Classnotes for Thursday September 30|Thursday]], [[10-327/Class Photo|Class Photo]], [[10-327/Homework Assignment 2|HW2]], [[10-327/Homework Assignment 2 Solutions|HW2 Solutions]]
|- align=left
|align=center|4
|Oct 4
|[[10-327/Classnotes for Monday October 4|Monday]] - the axiom of choice and infinite product spaces, [[10-327/Classnotes for Thursday October 7|Thursday]] - the box and the product topologies, metric spaces, [[10-327/Homework Assignment 3|HW3]], [[10-327/Homework Assignment 3 Solutions|HW3 Solutions]]
|- align=left
|align=center|5
|Oct 11
|Monday is Thanksgiving. [[10-327/Classnotes for Thursday October 14|Thursday]] - metric spaces, sequencial closures, various products. Final exam's date announced on Friday.
|- align=left
|align=center|6
|Oct 18
|[[10-327/Classnotes for Monday October 18|Monday]] - connectedness in <math>{\mathbb R}</math>, [[10-327/Homework Assignment 4|HW4]], [[10-327/Homework Assignment 4 Solutions|HW4 Solutions]], [[10-327/Classnotes for Thursday October 21|Thursday]] - connectedness, path-connectedness and products
|- align=left
|align=center|7
|Oct 25
|[[10-327/Classnotes for Monday October 25|Monday]] - compactness of <math>[0,1]</math>, [[10-327/Term Test|Term Test]] on Thursday
|- align=left
|align=center|8
|Nov 1
|[[10-327/Classnotes for Monday November 1|Monday]] - compact is closed and bounded, maximal values, [[10-327/Homework Assignment 5|HW5]], Wednesday was the last date to drop this course, [[10-327/Classnotes for Thursday November 4|Thursday]] - compactness of products and in metric spaces, the FIP
|- align=left
|align=center|9
|Nov 8
|Monday-Tuesday is Fall Break, [[10-327/Classnotes for Thursday November 11|Thursday]] - Tychonoff and a taste of Stone-Cech, [[10-327/Homework Assignment 6|HW6]]
|- align=left
|align=center|10
|Nov 15
|[[10-327/Classnotes for Monday November 15|Monday]] - generalized limits, [[10-327/Classnotes for Thursday November 18|Thursday]] - Normal spaces and Urysohn's lemma, [[10-327/Homework Assignment 7|HW7]]
|- align=left
|align=center|11
|Nov 22
|[[10-327/Classnotes for Monday November 22|Monday]] - <math>T_{3.5}</math> and <math>I^A</math>, [[10-327/Classnotes for Thursday November 25|Thursday]] - Tietze's theorem
|- align=left
|align=center|12
|Nov 29
|[[10-327/Classnotes for Monday November 29|Monday]] - compactness in metric spaces, [[10-327/Homework Assignment 8|HW8]], [[10-327/Classnotes for Thursday December 2|Thursday]] - completeness and compactness
|- align=left
|align=center|13
|Dec 6
|[[10-327/Classnotes for Monday December 6|Monday]] - Baire spaces and no-where differentiable functions, [[10-327/Classnotes for Wednesday December 8|Wednesday]] - Hilbert's 13th problem; also see [[December 2010 Schedule]]
|- align=left
|align=center|R
|Dec 13
|See [[December 2010 Schedule]]
|- align=left
|align=center|F
|Dec 20
|Final exam, Monday December 20, 2PM-5PM, at BR200
|- align=left
|colspan=3 align=center|[[10-327/Register of Good Deeds|Register of Good Deeds]] / [[10-327/To Do|To Do List]]
|- align=left
|colspan=3 align=center|[[Image:10-327-ClassPhoto.jpg|310px]] [[10-327/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-327-Splash.png|310px]] See {{Home Link|Talks/Fields-0911/|Hilbert's 13th}}
|}

Notes for Topology-101129/0:51:00

2010-12-04T05:34:58Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-150200.jpg Blackboard shot 9]: End of proof.

Notes for Topology-101129/0:45:35

2010-12-04T05:33:48Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-150153.jpg Blackboard shot 8]: Proof of 3<math>\Rightarrow</math>totally bounded

Notes for Topology-101129/0:28:10

2010-12-04T05:33:26Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-144510.jpg Blackboard shot 6]: Proof of 2<math>\Rightarrow</math>3 (infinite case)

Notes for Topology-101129/0:22:00

2010-12-04T05:33:09Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-144504.jpg Blackboard shot 5]: Proof of 2<math>\Rightarrow</math>3 (finite case)

Notes for Topology-101129/0:10:26

2010-12-04T05:32:33Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-142555.jpg Blackboard shot 3]: Proof of 1<math>\Rightarrow</math>2 (see [http://katlas.math.toronto.edu/drorbn/images/4/40/10-327-CompactnessInMetricSpaces.png Handout])

Notes for Topology-101129/0:51:00

2010-12-04T05:31:01Z

Anne.d:

[Blackboard shot 9]: End of proof.

Notes for Topology-101129/0:45:35

2010-12-04T05:30:42Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-150153.jpg Blackboard shot 8]: 3<math>\Rightarrow</math>totally bounded

Notes for Topology-101129/0:36:40

2010-12-04T05:30:32Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-150147.jpg Blackboard shot 7]: Totally bounded<math>\Rightarrow</math>bounded

Notes for Topology-101129/0:28:10

2010-12-04T05:30:26Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-144510.jpg Blackboard shot 6]: 2<math>\Rightarrow</math>3 (infinite case)

Notes for Topology-101129/0:22:00

2010-12-04T05:30:19Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-144504.jpg Blackboard shot 5]: 2<math>\Rightarrow</math>3 (finite case)

Notes for Topology-101129/0:17:16

2010-12-04T05:30:05Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-144458.jpg Blackboard shot 4]: Sequential Compactness. Example.

Notes for Topology-101129/0:10:26

2010-12-04T05:29:19Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-142555.jpg Blackboard shot 3]: Proof of 1<math>\Rightarrow</math>2

Notes for Topology-101129/0:05:25

2010-12-04T05:28:57Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-142550.jpg Blackboard shot 2]: Limit point compactness.

Notes for Topology-101129/0:00:07

2010-12-04T05:28:49Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101129-142544.jpg Blackboard shot 1]: Riddle, Announcements. 5 equivalent conditions for compactness in metric following the [http://katlas.math.toronto.edu/drorbn/images/4/40/10-327-CompactnessInMetricSpaces.png Handout].

Notes for Topology-101115/0:44:00

2010-11-18T01:44:29Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-150131.jpg Blackboard shot 7:] <math>\mbox{Lim}_\mu </math> generalizes properties of ordinary limit; <math>\mu</math> exists but is not useful.

Notes for Topology-101115/0:44:00

2010-11-18T01:43:44Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-150131.jpg Blackboard shot 8:] <math>\mbox{Lim}_\mu </math> generalizes properties of ordinary limit; <math>\mu</math> exists but is not useful.

Notes for Topology-101115/0:38:50

2010-11-18T01:43:19Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-145315.jpg Blackboard shot 6:] Proof continued.

Notes for Topology-101115/0:44:00

2010-11-18T01:41:35Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-150131.jpg Blackboard shot 8:] <math>\mu</math> exists but is not useful.

Notes for Topology-101115/0:38:50

2010-11-18T01:41:01Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-145315.jpg Blackboard shot 6:] Proof continued. <math>\mbox{Lim}_\mu </math> generalizes properties of ordinary limit.

Notes for Topology-101115/0:36:00

2010-11-18T01:40:49Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-145308.jpg Blackboard shot 5:] Proof continued.

Notes for Topology-101115/0:27:48

2010-11-18T01:40:07Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-145303.jpg Blackboard shot 4:] Convergent sequence Lemma. Proof of <math>\mbox{Lim}_\mu b_{j}=\lim b_{j}</math> if <math>b_{j}</math> converges.

Notes for Topology-101115/0:22:23

2010-11-18T01:39:45Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-143652.jpg Blackboard shot 3:] Claim: <math>\mu \in \beta\N\backslash\N</math> is too good. Everything converges. <math>\mbox{Lim}_\mu </math> is well-behaved.

Notes for Topology-101115/0:06:28

2010-11-18T01:39:20Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-143646.jpg Blackboard shot 2:] What does <math>\beta\N</math> look like in <math>C</math>? Prop: Every bndd seq has a unique cont. extension.

Notes for Topology-101115/0:00:10

2010-11-18T01:37:58Z

Anne.d:

[http://katlas.math.toronto.edu/drorbn/bbs/show?shot=10_327-101115-143641.jpg Blackboard shot 1:] Riddle; Stone-Cech Reminders

10-327/Homework Assignment 3

2010-11-12T02:44:53Z

Anne.d:

{{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 world, 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)
** Part of the reason I've decided to extend the deadline for this assignment is to have more time to talk about it in class. [[User:Drorbn|Drorbn]] 06:53, 14 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 <math>\ell^2</math> topology. Dror could you possibly explain on this for a little bit please? -Kai [[User:Xwbdsb|Xwbdsb]] 00:18, 13 October 2010 (EDT)
** That may be a good topic for a few minutes in class. If the question will be raised I'll try to give a hint. [[User:Drorbn|Drorbn]] 06:53, 14 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 answer 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
*** I completely agree with you. its not that you are dumb but it is really hard especially the last one. Who by any chance know how open balls of l2 topology looks like in either R^infinity or Hilbert cube? I have no idea and cannot conceptualize what is going on. [[User:Xwbdsb|Xwbdsb]] 01:35, 14 October 2010 (EDT)

* I would like to share a hint from Dror with everybody who cares. This thing puzzled me so that I couldn't figure a lot of things out. We can think of R^infinity as a space with all the sequences eventually end with 0. You might think that a sequence can only have finitely many non-zero terms. You are right. But philosophically does finite really mean "finite"? The hint here is that you can have a sequence that has as many non zero terms as you want in R infinity as long as it is finite... !!really as many non-zero terms as you want!! That just means that they sort of behave similarly to those sequences that have infinitely many non-zero terms. But the subtlety is there we cannot misunderstand it. -Kai[[User:Xwbdsb|Xwbdsb]] 00:40, 15 October 2010 (EDT)

*Anybody wants to share the happiness of solving HW3? This HW is so tough.. It really took me 11 pages to solve 8 which I never thought I could... We should definitely help each other out here on Wiki because the assignments are getting harder and harder. I would like to be the first person to help here on Wiki... -Kai [[User:Xwbdsb|Xwbdsb]] 00:40, 16 October 2010 (EDT)

10-327/Homework Assignment 2

2010-11-12T02:43:37Z

Anne.d:

{{10-327/Navigation}}

===Reading===
Read sections 17 through 21 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 22 through 24, 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, 8, 13, 14, 19abc, 19d, 21 on pages 101-102, and problems 7a, 7b, 8, 9ab, 9c, 13 on pages 111-112.

===Due date===
This assignment is due at the end of class on Thursday, October 7, 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}}

===Remark on the Due Date===
*Dear Professor Bar-Natan, October 5 seems like a Tuesday. Do you mean October 7, 2010? Thanks! [[User:Fzhao|Fzhao]] 23:42, 30 September 2010 (EDT)Frank
**I stand corrected. [[User:Drorbn|Drorbn]] 06:33, 1 October 2010 (EDT)

===Questions===
*Hi, I have a quick question. In the last question on the assignment that is being marked, what does it mean for one function to "uniquely determine" another. Sorry, I have just never heard that terminology before. - Jdw
** It means that any two functions with the property stated in the question are actually the same. [[User:Drorbn|Drorbn]] 07:19, 2 October 2010 (EDT)
*[[User:Xwbdsb|Xwbdsb]] 00:39, 2 October 2010 (EDT) I have a question about problem 13 on page 101. What does <math>x\times x</math> mean when <math>x</math> is an element in <math>X</math>? Does the author mean the ordered pair <math>(x,x)</math>? And we assume that we put product topology on <math>X \times X</math>? -Kai
** Yes and yes. [[User:Drorbn|Drorbn]] 07:19, 2 October 2010 (EDT)
*** I found a way to approach this problem but I am not sure about the technicality. <math>X \times X</math> is Hausdorff. We take any point in <math>\Delta</math> complement. So we can separate it from any point in <math>\Delta</math>. But to separate it from the entire <math>\Delta</math> we need to get the intersection of all its open nbds. Will that still be a valid open nbd? -Kai [[User:Xwbdsb|Xwbdsb]] 11:29, 2 October 2010 (EDT)
**** An arbitrary intersection of open sets is not necessarily open. This I'll say, but beyond this, it is your problem to solve. [[User:Drorbn|Drorbn]] 16:21, 2 October 2010 (EDT)

10-327/Homework Assignment 1

2010-11-12T02:42:55Z

Anne.d:

{{10-327/Navigation}}

'''Read''' sections 12 through 17 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 18 through 22, just to get a feel for the future.

'''Solve and submit the following problems.''' In Munkres' book, problems 4 and 8 on pages 83-84, problems 4 and 8 on page 92, problems 3 and 4 on page 100, and, for extra credit, the following problem:

''Problem.'' Let <math>X</math> and <math>Y</math> be topological spaces and let <math>A\subset X</math> and <math>B\subset Y</math> be subsets thereof. Using only the definitions in terms of continuity of certain functions, show that the topology induced on <math>A\times B</math> as a subset of the product <math>X\times Y</math> is equal to the topology induced on it as a product of subsets of <math>X</math> and of <math>Y</math>. You are allowed to use the fact that two topologies <math>{\mathcal T}_1</math> and <math>{\mathcal T}_2</math> on some set <math>W</math> are equal if and only if the identity map regarded as a map from <math>(W, {\mathcal T}_1)</math> to <math>(W, {\mathcal T}_2)</math> is a homeomorphism. Words like "open sets" and "basis for a topology" are not allowed in your proof.

'''Due date.''' This assignment is due at the end of class on Thursday, September 30, 2010.

'''Note on Question 8, Page 92.''' (Added 8:00AM, September 27). One should think that "describe" for verbal things is like "simplify" for formula-things. The topologies in question were given by a verbal description; the content of the question is that you should be giving a simpler one, and the best is if it is of the form "the topology in question is the trivial topology", or something like that. Note that the resulting topology may also depend on the direction of the line <math>L</math>, so you may wish to divide your answer into parts depending on that direction.

{{Template:10-327:Dror/Students Divider}}
* Dror do you mean question 3 and 4 on page 100? There is no question 3 and 4 on page 101. -Kai
** Sorry and thanks for the correction, indeed I meant page 100. (BTW, next time you sign a wiki submission, use "<nowiki>~~~~</nowiki>" (four "tilde" symbols) and see what it does). [[User:Drorbn|Drorbn]] 18:03, 26 September 2010 (EDT)

10-327/Homework Assignment 3 Solutions

2010-11-12T02:40:22Z

Anne.d:

{{10-327/Navigation}}

=== Solution ===
Sorry the midterm took me a lot of time to prepare here I attach the solution for HW3.
Notice that this time I have a much larger scanner so I scan 2 pages per time, but i write things front and back so is a bit hard for me to scan so please keep track of the page number at the bottom thanks. Page number goes like 1,3 2,4 5,7,6,8...etc -Kai [[User:Xwbdsb|Xwbdsb]] 16:33, 28 October 2010 (EDT)

[http://katlas.math.toronto.edu/drorbn/images/1/1e/10-327hw3-1.jpg page1]
[http://katlas.math.toronto.edu/drorbn/images/e/e0/10-327hw3-2.jpg page2]
[http://katlas.math.toronto.edu/drorbn/images/5/53/10-327hw3-3.jpg page3]
[http://katlas.math.toronto.edu/drorbn/images/a/a4/10-327hw3-4.jpg page4]
[http://katlas.math.toronto.edu/drorbn/images/e/e6/10-327hw3-5.jpg page5]
[http://katlas.math.toronto.edu/drorbn/images/0/00/10-327hw3-6.jpg page6]
[http://katlas.math.toronto.edu/drorbn/images/7/7d/10-327hw3-7(modified).jpg page7]
[http://katlas.math.toronto.edu/drorbn/images/0/0b/10-327hw3-8.jpg page8]
[http://katlas.math.toronto.edu/drorbn/images/f/fc/10-327hw3-9.jpg page9]
[http://katlas.math.toronto.edu/drorbn/images/b/b9/10-327hw3-10.jpg page10]
[http://katlas.math.toronto.edu/drorbn/images/c/c6/10-327hw3-11.jpg page11]
[http://katlas.math.toronto.edu/drorbn/images/7/72/10-327hw3-12.jpg page12]

From now on I will just scan my HW like that which will give rid of the hassle of finding a 3rd party to upload files.-Kai

10-327/Homework Assignment 2 Solutions

2010-11-12T02:39:22Z

Anne.d:

{{10-327/Navigation}}

===Solution===

As promised, I provide my solution to HW2 here for any one of you might care. -Kai
http://www.2shared.com/document/DVGaHsqO/HW2sol.html

Also try this link if the previous one does not work.
http://www.megaupload.com/?d=QHKY2FVW

Here are some sketchy solutions to some extra problems. They are not complete and I still could not solve some of them.
So of course be skeptical. -Kai http://www.2shared.com/document/eomly5cw/HW2extraproblems.html

Also try this link http://www.megaupload.com/?d=VUUX4E8G -Kai [[User:Xwbdsb|Xwbdsb]] 09:25, 15 October 2010 (EDT)

10-327/Homework Assignment 1 Solutions

2010-11-12T02:39:03Z

Anne.d:

{{10-327/Navigation}}

===Solution===
As promised, I provide my solution to HW1 here for any one of you might care. I found a 3rd party website because I Couldn't upload pdf documents. -Kai
http://www.2shared.com/document/9taHAAOM/HW1sol.html

Or try this link if the above link does not work. http://www.megaupload.com/?d=9ZBVKTS5 -Kai[[User:Xwbdsb|Xwbdsb]] 09:17, 15 October 2010 (EDT)

10-327/Homework Assignment 2 Solutions

2010-11-12T02:37:38Z

Anne.d:

===Solution===

As promised, I provide my solution to HW2 here for any one of you might care. -Kai
http://www.2shared.com/document/DVGaHsqO/HW2sol.html

Also try this link if the previous one does not work.
http://www.megaupload.com/?d=QHKY2FVW

Here are some sketchy solutions to some extra problems. They are not complete and I still could not solve some of them.
So of course be skeptical. -Kai http://www.2shared.com/document/eomly5cw/HW2extraproblems.html

Also try this link http://www.megaupload.com/?d=VUUX4E8G -Kai [[User:Xwbdsb|Xwbdsb]] 09:25, 15 October 2010 (EDT)