Drorbn - User contributions [en]

10-327/Classnotes for Monday December 6

2010-12-21T00:14:04Z

Xwbdsb:

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

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

*'''Question.''' The fact that the metric space of real-valued functions on the unit interval with uniform metric is complete uses the fact that [0,1] is compact right? If the function space is defined on a non-compact topological space is that necessarily complete?... -Kai [[User:Xwbdsb|Xwbdsb]] 00:01, 20 December 2010 (EST)
** No, the compactness of <math>[0,1]</math> is not used. As we said in class, if <math>(f_n)</math> is Cauchy in the uniform metric, then for any <math>x</math>, the sequence <math>(f_n(x))</math> is Cauchy in <math>{\mathbb R}</math>, so it has a limit. Call that limit <math>f(x)</math>; it is not hard to show that <math>f</math> is continuous and that <math>f_n\to f</math>. Theorem 43.6 in Munkres is a slight generalization of this. [[User:Drorbn|Drorbn]] 07:12, 20 December 2010 (EST)
Thanks Dror.

Everybody good luck on the exam!-Kai

Great course! Thank you very much for all your help Dror and all the classmates in this class.
-Kai

10-327/Classnotes for Monday December 6

2010-12-20T15:45:45Z

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10-327/Classnotes for Monday December 6

2010-12-20T05:01:31Z

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10-327/Homework Assignment 8 Solutions

2010-12-19T22:53:04Z

Xwbdsb:

{{10-327/Navigation}}
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]
[http://katlas.math.toronto.edu/drorbn/images/b/b9/10-327a802.JPG page2]
[http://katlas.math.toronto.edu/drorbn/images/f/f0/10-327a803.JPG page3]
[http://katlas.math.toronto.edu/drorbn/images/a/a9/10-327a804.JPG page4]
[http://katlas.math.toronto.edu/drorbn/images/6/61/10-327a805.JPG page5]
[http://katlas.math.toronto.edu/drorbn/images/3/32/10-327a806.JPG page6]
[http://katlas.math.toronto.edu/drorbn/images/d/d0/10-327a807.JPG page7]
[http://katlas.math.toronto.edu/drorbn/images/4/4d/10-327a808.JPG page8]
[http://katlas.math.toronto.edu/drorbn/images/a/a2/10-327a809.JPG page9]
[http://katlas.math.toronto.edu/drorbn/images/6/6c/10-327a810.JPG page10]
[http://katlas.math.toronto.edu/drorbn/images/1/1d/10-327a811.JPG page11]
[http://katlas.math.toronto.edu/drorbn/images/f/fb/10-327a812.JPG page12]
[http://katlas.math.toronto.edu/drorbn/images/b/b0/10-327a813.JPG page13]
[http://katlas.math.toronto.edu/drorbn/images/1/1d/10-327a814.JPG page14]
[http://katlas.math.toronto.edu/drorbn/images/6/62/10-327a815.JPG page15]
[http://katlas.math.toronto.edu/drorbn/images/9/9c/10-327a816.JPG page16]

-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?

-The resulting product space is complete only if you put that weird metric that induces the product topology. When the space is not metric we cannot talk about completeness. Converse depends on what kind of metric you define on the projection? countable infinite product is done in the book. It is complete with that weird metric induced from metrics from all the spaces.-Kai

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]

Talk:10-327/Register of Good Deeds

2010-12-19T13:02:42Z

Xwbdsb:

*10-327/Kai Yang 997712756
1.Assignment solutions 1-8, term test solutions.

2.For lecture 3. Complete proof for the equality of three topologies generated by basis B. And solution to the four exercises.

3.For lecture 4. Some more illustration on Uniqueness of the product topology satisfying condition 1&2. Complete proof of the subspace topology is the unique topology satisfying condition 1&2. Proof for a couple of claims: The product topology on R_std and R_std is the standard topology on R^2 and subspace topology on Z as a subspace of Y which is a subspace of Z is the same as the subspace topology on Z as a subspace of X, where Y is a subspace of X.

4.Some extra problem (not to be handed in) for HW2.

5.Annotated a few lecture videos.

6.A few more solutions to questions about metric spaces with illustrations.

7.A picture to summarize properties/thms about T3/3.5/4/4.5 spaces.

10-327/Classnotes for Thursday November 25

2010-12-19T12:59:52Z

Xwbdsb:

{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101125-142103.jpg}}.

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Here is a lecture note for today:

[http://katlas.org/drorbn/images/3/38/10-327_Nov_25_LecNotes.pdf Lecture Nov 25]

=== Question ===
Question. The first half of Tietze's theorem isn't very surprising as a limiting process of approximations.
But the second half is just like a magic? I don't understand what has been implicitly used here. The "boundedness"
property only depends on the metric we define on a set and it does not have anything to do with topology.
We are linking R with (-1,1) with a homeomorphism which is completely not metric-related. And suddenly all the unbounded
cts functions all become bounded cts functions?......What has been used here? Did we implicitly redefined the metric?
Why it works out so smoothly just like a magic trick?...

-Kai

Kai - your question is too open-ended to have an answer that fits in a few minutes of typing, so I'd rather answer it in person, if you come to my office hours. [[User:Drorbn|Drorbn]] 16:39, 6 December 2010 (EST)

-Picture

One picture summary of what you should know about regular/completely regular/normal/completely normal spaces. -Kai[[User:Xwbdsb|Xwbdsb]] 07:59, 19 December 2010 (EST)
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_review.JPG

File:10-327 review.JPG

2010-12-19T12:59:29Z

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10-327/Classnotes for Thursday November 4

2010-12-19T12:43:28Z

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{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101104-142342.jpg}}.

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* Question: Regarding the proof of the Lebesque Number Lemma, let T(x) be the function we were working with in the proof. I am confused with how we reached the conclusion that if d(x,y)<E, then T(y)>= T(x) - E. I know that it was said that this is just an application of the triangle inequality, but I am having a bit of trouble seeing that. Hopefully someone can make this point a bit clearer for me. Thanks! Jason.
** If you could find a ball of radius 7 around <math>x</math> which fits inside some set <math>U</math>, and you move <math>x</math> just a 1 unit away to <math>y</math>, then by the triangle inequality the ball of radius 6 around <math>y</math> is entirely contained inside the ball of radius 7 around <math>x</math> so it is entirely contained in <math>U</math>. [[User:Drorbn|Drorbn]] 18:37, 6 November 2010 (EDT)
***I have some doubts with Lebesgue number lemma too.. this delta(x) isn't a radius that we can fit a ball inside one of the U's. It is the supremum of all possible radius. Wouldn't that give us a problem? Don't we need to subtract some small positive value and then find a valid radius? I am not sure if there should be some technicality involved here.-Kai
And once we found this delta(x_0) we should divide by 2 so that delta(x)>delta(x_0) for all x in X? Because it is not necessarily true that we can find a ball of radius delta(x_0) around x_0 to fit into ball?-Kai
*I think you are correct, (if you follow the proof on the blackboard shots to the letter) as an example of what you are talking about take the open cover of [0,1] using sets of the form [0,x) for x<3/4 and (y,1] for y>1/4. Then <math>\Delta (x)\geq 1/4</math> and at 1/2 <math>\Delta (1/2)=1/4</math> but no ball of radius 1/4 around 1/2 fits inside any of the sets. So we will need to take a smaller value than <math>inf(\Delta (x))</math> as our value of <math>\delta_0</math> for Lebesgue's Lemma. Dividing by two as you suggest should work fine to fix this problem... Maybe it's just me, but in proof's like this I always feel the urge to divide the final answer by 2, just in case I mixed up some some strict inequality, with a non-strict one somewhere - [[Johnfleming|John]]
**Thanks John very nice example. Can you also help me with this question? "delta(x) isn't a radius that we can fit a ball inside one of the U's. It is the supremum of all possible radius. Wouldn't that give us a problem in showing delta(y)>=delta(x)-epsilon when d(x,y)<epsilon?"-Kai
***I don't think there is any problem with this step.
Without loss of generality <math> \epsilon < \Delta (x)</math> otherwise the condition holds vacuously.
Suppose we construct a ball of radius r such that <math> r > \epsilon </math> around x so that it is a subset of some U in the cover, then we can construct a ball of radius <math>r - \epsilon</math> around y such that this ball is also in U. So <math> \Delta (y) \geq r-\epsilon </math> for all r such that a ball of radius r exists around x as in the proof. Which implies <math> \Delta (y) \geq sup(r) - \epsilon </math> implies <math> \Delta (y) \geq \Delta (x) - \epsilon </math>. Hopefully this makes sense and works - [[Johnfleming|John]]
-Yes. Makes sense. Nice and concise thanks! -Kai

10-327/Homework Assignment 8 Solutions

2010-12-19T12:32:51Z

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10-327/Classnotes for Thursday November 4

2010-12-18T23:13:31Z

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10-327/Classnotes for Thursday November 4

2010-12-18T05:15:38Z

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10-327/Classnotes for Thursday November 4

2010-12-18T04:51:35Z

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10-327/Homework Assignment 8

2010-12-06T14:48:29Z

Xwbdsb: /* Discussion */

{{10-327/Navigation}}
===Change in Plans===

Time to rest, a bit. For the lack of fun questions, I've decided to postpone this assignment a bit, and to cancel assignment number 9. So this assignment will only be written on Monday November 29, and will be due on the following Monday, December 6.

===Reading===
'''Read''' sections 35, 43, and 45 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''' section 48 and 49, just to get a feel for the future. For fun alone, read also section 44.

===Doing===
Solve and submit the following problems from Munkres' book:
* Problems 2, 5a on pages 223-224.
* Problems 1, 4, 5 on page 270 (the relevant material will be covered on Thursday December 2).
* Problem 1 on page 280 (same comment).

In addition, solve and submit the following problem:

'''Problem IX.''' Show that every metric space <math>X</math> can be embedded in the cube <math>I^X</math>.

''Hint.'' Given a point <math>x\in X</math>, what real-valued function on <math>X</math> immediately comes to mind?

===Due date===
This assignment is due at the end of class on Wednesday, December 8, 2010.

{{Template:10-327:Dror/Students Divider}}
===Discussion===
* Then how do we count the homework in the final grade? I remember the old rule is to select the best 7 assignments from the total of 9. - Jolin
** The rule was "worst two assignments will not count", see [[10-327/About This Class]]. [[User:Drorbn|Drorbn]] 05:40, 27 November 2010 (EST)

Question.
For question 4. It is to show the intersection is nonempty by creating a Cauchy sequence from these A_n's. But I can't think of a way to prove the other direction. Are we supposed to create a nested sequence of closed sets for every Cauchy sequence? Then how do we ensure closeness?
-Kai

Take the closure of each set, as your set.
-Jdw
Yea thanks Jason. It works because the diam of the closure of a set is the same as the set in the metric topology. -Kai

10-327/Homework Assignment 8

2010-12-06T04:22:53Z

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10-327/Classnotes for Thursday November 25

2010-12-05T18:12:53Z

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10-327/Homework Assignment 7

2010-12-04T18:16:33Z

Xwbdsb: /* Solution */

{{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)

----
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)

===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 7

2010-12-04T18:14:13Z

Xwbdsb: /* Due date */

{{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)

----
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)

===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/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]
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[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]

10-327/Homework Assignment 6

2010-12-04T18:13:22Z

Xwbdsb: /* Solution */

{{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)

===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]
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[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]
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[http://katlas.math.toronto.edu/drorbn/images/e/e1/10-327a613.JPG page13]

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