http://drorbn.net/api.php?action=feedcontributions&user=Johnfleming&feedformat=atomDrorbn - User contributions [en]2024-03-29T13:08:07ZUser contributionsMediaWiki 1.21.1http://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_November_410-327/Classnotes for Thursday November 42010-12-19T02:35:46Z<p>Johnfleming: </p>
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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.<br />
** 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)<br />
***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<br />
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<br />
*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]]<br />
**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<br />
***I don't think there is any problem with this step. <br />
Without loss of generality <math> \epsilon < \Delta (x)</math> otherwise the condition holds vacuously.<br />
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]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_November_410-327/Classnotes for Thursday November 42010-12-18T21:22:15Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
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See some blackboard shots at {{BBS Link|10_327-101104-142342.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
* 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.<br />
** 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)<br />
***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<br />
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<br />
*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]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Navigation10-327/Navigation2010-12-14T02:04:37Z<p>Johnfleming: </p>
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<div><noinclude>Back to [[10-327]].<br/></noinclude><br />
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"<br />
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|colspan=3 style="color: red;"|'''Additions to the MAT 327 web site will count towards [[10-327/Register of Good Deeds|good deed points]] until Sunday December 19 at 5PM'''<br />
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!#<br />
!Week of...<br />
!Notes and Links<br />
|- align=left<br />
|align=center|1<br />
|Sep 13<br />
|[[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.<br />
|- align=left<br />
|align=center|2<br />
|Sep 20<br />
|[[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]]<br />
|- align=left<br />
|align=center|3<br />
|Sep 27<br />
|[[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]]<br />
|- align=left<br />
|align=center|4<br />
|Oct 4<br />
|[[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]]<br />
|- align=left<br />
|align=center|5<br />
|Oct 11<br />
|Monday is Thanksgiving. [[10-327/Classnotes for Thursday October 14|Thursday]] - metric spaces, sequencial closures, various products. Final exam's date announced on Friday.<br />
|- align=left<br />
|align=center|6<br />
|Oct 18<br />
|[[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<br />
|- align=left<br />
|align=center|7<br />
|Oct 25<br />
|[[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]]<br />
|- align=left<br />
|align=center|8<br />
|Nov 1<br />
|[[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<br />
|- align=left<br />
|align=center|9<br />
|Nov 8<br />
|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]]<br />
|- align=left<br />
|align=center|10<br />
|Nov 15<br />
|[[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]]<br />
|- align=left<br />
|align=center|11<br />
|Nov 22<br />
|[[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<br />
|- align=left<br />
|align=center|12<br />
|Nov 29<br />
|[[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<br />
|- align=left<br />
|align=center|13<br />
|Dec 6<br />
|[[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]]<br />
|- align=left<br />
|align=center|R<br />
|Dec 13<br />
|See [[December 2010 Schedule]]<br />
|- align=left<br />
|align=center|F<br />
|Dec 20<br />
|Final exam, Monday December 20, 2PM-5PM, at BR200<br />
|- align=left<br />
|colspan=3 align=center|[[10-327/Register of Good Deeds|Register of Good Deeds]]<br />
|- align=left<br />
|colspan=3 align=center|[[Image:10-327-ClassPhoto.jpg|310px]]<br/>[[10-327/Class Photo|Add your name / see who's in!]]<br />
|- align=left<br />
|colspan=3 align=center|[[Image:10-327-Splash.png|310px]]<br/>See {{Home Link|Talks/Fields-0911/|Hilbert's 13th}}<br />
|}</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Solution_to_configuration_of_a_3_legged_spider10-327/Solution to configuration of a 3 legged spider2010-11-25T01:23:54Z<p>Johnfleming: </p>
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<div>[[Image:10-327-3_legged_spider.png]]<br />
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It's a sphere.<br />
Divide the sphere into 8 section, and have the middle joint denote where on each section a configuration relates to.<br />
Then label each section by how the joints are bent (up or down, left or right). If the middle joint is at the boundary of where it can move then one of the joints must be straight (both up and down) and can move from one section to another by small permutations. <br />
Hopefully this makes sense.<br />
I believe you can also prove it by looking at the "level sets" of the configuration, by fixing one joint and looking at the "sub"-configuration space. (Like how the sphere was done in class)<br />
-[[User:Johnfleming|John]]</div>Johnfleminghttp://drorbn.net/index.php?title=File:10-327-3_legged_spider.pngFile:10-327-3 legged spider.png2010-11-25T01:09:14Z<p>Johnfleming: Configuration space of a three legged spider.</p>
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<div>Configuration space of a three legged spider.</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Monday_November_2210-327/Classnotes for Monday November 222010-11-25T00:58:50Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
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See some blackboard shots at {{BBS Link|10_327-101122-143551.jpg}}.<br />
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{{Template:10-327:Dror/Students Divider}}<br />
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===Riddle Along===<br />
What is the configuration space of a 3 legged spider<br />
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[[10-327/Solution to configuration of a 3 legged spider]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_November_1810-327/Classnotes for Thursday November 182010-11-20T18:49:36Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101118-142430.jpg}} (unfortunately some blackboards from the middle of the proof or Urysohn's lemma got forgotten).<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
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Here are some lecture notes:<br />
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[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Image-10-327_lecpt1.jpg Lecture part 1]<br />
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[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_lecpt2.jpg Lecture part 2]<br />
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[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_lecpt3.jpg Lecture part 3]<br />
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[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_lecpt4.jpg Lecture part 4]<br />
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'''Questions by Kai [[User:Xwbdsb|Xwbdsb]] 21:26, 19 November 2010 (EST):'''<br />
<br />
# I have concern about this "Adding T_1 thing". When we are proving the Urysohn's Lemma, we proved that if X is a T.S. and A,B are any two disjoint closed sets in X, they can be separated by two disjoint open sets in X iff they can be separated by a continuous function. So there is no T_1 axiom involved in any of these two proofs. So Urysohn's Lemma is something more general than T_4 iff T_4.5 right?<br />
#* You are probably right and Urysohn applies even without <math>T_1</math>. Yet we want <math>T1</math> all else <math>T_{4.5}</math> does not imply <math>T_{3.5}</math> and below, and we will need <math>T_{3.5}</math> very soon. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
# I just want to also confirm the definition of T_4 and T_4.5: T_4 is T_1 + separation by two disjoint open sets for any two disjoint closed sets and T_4.5 is T_1 + separation by a continuous function for any two disjoint closed sets?<br />
#* Indeed so. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
# The only purpose of this manually adding T_1 thing is that so that T_4 and T_4.5 could imply T_3 T_2 T_1 right?<br />
#* Indeed. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
# Dror you said before that induction can go up to finite n but why in this case induction will work for a countably infinite set of elements?<br />
#* We are never making a statement about an infinite set of rational numbers directly through induction. We are only constructing something for the <math>n</math>'th rational, for each <math>n</math>. We then look at the totality of these constructions, but that's already ''outside'' the induction. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
**So you mean if properly used, mathematical induction could prove things involve countably infinite objects right? Just like in this case. -Kai[[User:Xwbdsb|Xwbdsb]] 11:32, 20 November 2010 (EST)<br />
***Induction can be hard to understand, what induction bascially says is; If Property P holds for 1, and P holds for k implies P holds for k+1 then P holds for all natural numbers. So the property holds "with respect to" an a infinite coutable set (<math>\mathbb{N}</math>) but this does not mean the property holds "for" infinite cardinals. <br />
So something like, The finite intersection of open sets is open could use an inductive prove<br />
the intersection of one open set is open and the intersection of k open sets is open implies the intersection of k+1 open sets is open, then the intersection of any natural number of open sets is open. But of course infinite intersections of open sets are not necessarily open. So in this case there are an infinite number of ways to take intersections, but we cannot use infinite intersections themselves.<br />
In the proof done in class, we basically said, take a set with the first n rational numbers (giving by some ordering we choose) that has the required construction then we prove we can construct something for the set with the first n+1 rational numbers. So we have the construction that works for any set that has the cardinality of a natural number, but the construction does not work for a set with an infinite cardinality.<br />
<br />
I don't know if this helps, it took me a while to grasp this idea and it's how I think about it. -[[Johnfleming|John]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_November_1810-327/Classnotes for Thursday November 182010-11-20T18:49:12Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101118-142430.jpg}} (unfortunately some blackboards from the middle of the proof or Urysohn's lemma got forgotten).<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes:<br />
<br />
[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Image-10-327_lecpt1.jpg Lecture part 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_lecpt2.jpg Lecture part 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_lecpt3.jpg Lecture part 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327_lecpt4.jpg Lecture part 4]<br />
<br />
'''Questions by Kai [[User:Xwbdsb|Xwbdsb]] 21:26, 19 November 2010 (EST):'''<br />
<br />
# I have concern about this "Adding T_1 thing". When we are proving the Urysohn's Lemma, we proved that if X is a T.S. and A,B are any two disjoint closed sets in X, they can be separated by two disjoint open sets in X iff they can be separated by a continuous function. So there is no T_1 axiom involved in any of these two proofs. So Urysohn's Lemma is something more general than T_4 iff T_4.5 right?<br />
#* You are probably right and Urysohn applies even without <math>T_1</math>. Yet we want <math>T1</math> all else <math>T_{4.5}</math> does not imply <math>T_{3.5}</math> and below, and we will need <math>T_{3.5}</math> very soon. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
# I just want to also confirm the definition of T_4 and T_4.5: T_4 is T_1 + separation by two disjoint open sets for any two disjoint closed sets and T_4.5 is T_1 + separation by a continuous function for any two disjoint closed sets?<br />
#* Indeed so. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
# The only purpose of this manually adding T_1 thing is that so that T_4 and T_4.5 could imply T_3 T_2 T_1 right?<br />
#* Indeed. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
# Dror you said before that induction can go up to finite n but why in this case induction will work for a countably infinite set of elements?<br />
#* We are never making a statement about an infinite set of rational numbers directly through induction. We are only constructing something for the <math>n</math>'th rational, for each <math>n</math>. We then look at the totality of these constructions, but that's already ''outside'' the induction. [[User:Drorbn|Drorbn]] 06:02, 20 November 2010 (EST)<br />
**So you mean if properly used, mathematical induction could prove things involve countably infinite objects right? Just like in this case. -Kai[[User:Xwbdsb|Xwbdsb]] 11:32, 20 November 2010 (EST)<br />
***Induction can be hard to understand, what induction bascially says is; If Property P holds for 1, and P holds for k implies P holds for k+1 then P holds for all natural numbers. So the property holds "with respect to" an a infinite coutable set (<math>\mathbb{N}</math>) but this does not mean the property holds "for" infinite cardinals. <br />
So something like, The finite intersection of open sets is open could use an inductive prove<br />
the intersection of one open set is open and the intersection of k open sets is open implies the intersection of k+1 open sets is open, then the intersection of any natural number of open sets is open. But of course infinite intersections of open sets are not necessarily open. So in this case there are an infinite number of ways to take intersections, but we cannot use infinite intersections themselves.<br />
In the proof done in class, we basically said, take a set with the first n rational numbers (giving by some ordering we choose) that has the required construction then we prove we can construct something for the set with the first n+1 rational numbers. So we have the construction that works for any set that has the cardinality of a natural number, but the construction does not work for a set with an infinite cardinality.<br />
<br />
I don't know if this helps, it took me a while to grasp this idea and it's how I think about it.</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Homework_Assignment_610-327/Homework Assignment 62010-11-14T17:58:55Z<p>Johnfleming: /* Due date */</p>
<hr />
<div>{{10-327/Navigation}}<br />
===Reading===<br />
'''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.<br />
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===Doing===<br />
Solve and submit the following problems.<br />
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'''Problem 1.''' Problem 1 on page 235 of Munkres' book.<br />
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'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.<br />
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'''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>.<br />
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'''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>.<br />
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'''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>.<br />
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'''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>.<br />
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'''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:<br />
# <math>\mbox{SuperLim}(a_k)=\lim a_k</math>, if the sequence <math>a_k</math> is convergent.<br />
# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.<br />
# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.<br />
# <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>.<br />
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===Due date===<br />
This assignment is due at the end of class on Thursday, November 18, 2010.<br />
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{{Template:10-327:Dror/Students Divider}}<br />
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Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that?<br />
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?<br />
[[User:Xwbdsb|Xwbdsb]] 00:25, 13 November 2010 (EST)<br />
*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]]<br />
<br />
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)<br />
I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is<br />
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)<br />
<br />
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)<br />
<br />
* 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)<br />
<br />
* 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]]<br />
** Thanks! You are absolutely right. The question has been modified. [[User:Drorbn|Drorbn]] 15:23, 13 November 2010 (EST)<br />
<br />
----<br />
<br />
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?<br />
<br />
And what is the generalized definition of limit? I cannot find the definition anywhere.[[User:Xwbdsb|Xwbdsb]] 10:01, 14 November 2010 (EST)<br />
<br />
* 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)<br />
** 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]]<br />
<br />
*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)</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Homework_Assignment_610-327/Homework Assignment 62010-11-13T18:56:22Z<p>Johnfleming: /* Due date */</p>
<hr />
<div>{{10-327/Navigation}}<br />
===Reading===<br />
'''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.<br />
<br />
===Doing===<br />
Solve and submit the following problems.<br />
<br />
'''Problem 1.''' Problem 1 on page 235 of Munkres' book.<br />
<br />
'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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 3 properties:<br />
# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.<br />
# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.<br />
# <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>.<br />
<br />
===Due date===<br />
This assignment is due at the end of class on Thursday, November 18, 2010.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that?<br />
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?<br />
[[User:Xwbdsb|Xwbdsb]] 00:25, 13 November 2010 (EST)<br />
*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]]<br />
<br />
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)<br />
I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is<br />
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)<br />
<br />
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)<br />
<br />
* 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)<br />
<br />
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]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Homework_Assignment_610-327/Homework Assignment 62010-11-13T18:47:25Z<p>Johnfleming: /* Due date */</p>
<hr />
<div>{{10-327/Navigation}}<br />
===Reading===<br />
'''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.<br />
<br />
===Doing===<br />
Solve and submit the following problems.<br />
<br />
'''Problem 1.''' Problem 1 on page 235 of Munkres' book.<br />
<br />
'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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 3 properties:<br />
# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.<br />
# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.<br />
# <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>.<br />
<br />
===Due date===<br />
This assignment is due at the end of class on Thursday, November 18, 2010.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that?<br />
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?<br />
[[User:Xwbdsb|Xwbdsb]] 00:25, 13 November 2010 (EST)<br />
*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]]<br />
<br />
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)<br />
I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is<br />
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)<br />
<br />
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)<br />
<br />
* 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)<br />
<br />
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?-[[User:Johnfleming|John]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Homework_Assignment_610-327/Homework Assignment 62010-11-13T18:45:57Z<p>Johnfleming: /* Due date */</p>
<hr />
<div>{{10-327/Navigation}}<br />
===Reading===<br />
'''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.<br />
<br />
===Doing===<br />
Solve and submit the following problems.<br />
<br />
'''Problem 1.''' Problem 1 on page 235 of Munkres' book.<br />
<br />
'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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 3 properties:<br />
# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.<br />
# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.<br />
# <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>.<br />
<br />
===Due date===<br />
This assignment is due at the end of class on Thursday, November 18, 2010.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that?<br />
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?<br />
[[User:Xwbdsb|Xwbdsb]] 00:25, 13 November 2010 (EST)<br />
*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 <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]]<br />
<br />
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)<br />
I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is<br />
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)<br />
<br />
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)<br />
<br />
* 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)<br />
<br />
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?-[[User:Johnfleming|John]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Homework_Assignment_610-327/Homework Assignment 62010-11-13T18:44:28Z<p>Johnfleming: /* Due date */</p>
<hr />
<div>{{10-327/Navigation}}<br />
===Reading===<br />
'''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.<br />
<br />
===Doing===<br />
Solve and submit the following problems.<br />
<br />
'''Problem 1.''' Problem 1 on page 235 of Munkres' book.<br />
<br />
'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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>.<br />
<br />
'''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 3 properties:<br />
# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.<br />
# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.<br />
# <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>.<br />
<br />
===Due date===<br />
This assignment is due at the end of class on Thursday, November 18, 2010.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that?<br />
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?<br />
[[User:Xwbdsb|Xwbdsb]] 00:25, 13 November 2010 (EST)<br />
*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 <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.<br />
<br />
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)<br />
I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is<br />
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)<br />
<br />
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)<br />
<br />
* 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)<br />
<br />
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?</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_2110-327/Classnotes for Thursday October 212010-11-06T17:32:38Z<p>Johnfleming: /* Riddle Along */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101021-143325.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]<br />
<br />
==Riddle Along==<br />
Can you color <math>\mathbb{R}^2</math> with 4 colors such that no points with distance one are the same color?<br />
<br />
[[10-327/Solution to coloring R2]]<br />
*Are you asking us to solve the [http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]?<br />
** Only to give the relatively easy bound... [[User:Drorbn|Drorbn]] 08:58, 6 November 2010 (EDT)<br />
***Perhaps I am misreading the question, but according to the Wikipedia article and some quick conformational googling, it seems to be an open question as to whether or not it is possible with four colours. Three and Seven seem to be the easy bounds. Though, I suppose it is possible that four has recently been ruled out and the results require a bit more searching to find. Indecently, this was the Wikipedia's Mathmatics portal picture of the month which is how I stumbled across it. - [[User:Johnfleming|Johnfleming]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_2110-327/Classnotes for Thursday October 212010-11-06T17:23:27Z<p>Johnfleming: /* Riddle Along */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101021-143325.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]<br />
<br />
==Riddle Along==<br />
Can you color <math>\mathbb{R}^2</math> with 4 colors such that no points with distance one are the same color?<br />
<br />
[[10-327/Solution to coloring R2]]<br />
*Are you asking us to solve the [http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]?<br />
** Only to give the relatively easy bound... [[User:Drorbn|Drorbn]] 08:58, 6 November 2010 (EDT)<br />
***Perhaps I am misreading the question, but according to the Wikipedia article and some quick conformational googling, it seems to be an open question as to whether or not it is possible with four colours. Three and Seven seem to be the easy bound. Though, I suppose it is possible that four has recently been ruled out and the results require a bit more searching to find. Indecently, this was the Wikipedia's Mathmatics portal picture of the month which is how I stumbled across it. - [[User:Johnfleming|Johnfleming]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_2110-327/Classnotes for Thursday October 212010-11-06T00:36:08Z<p>Johnfleming: /* Riddle Along */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101021-143325.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]<br />
<br />
==Riddle Along==<br />
Can you color <math>\mathbb{R}^2</math> with 4 colors such that no points with distance one are the same color?<br />
<br />
[[10-327/Solution to coloring R2]]<br />
*Are you asking us to solve the [http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem Hadwiger-Nelson problem]?</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Homework_Assignment_410-327/Homework Assignment 42010-10-23T02:19:44Z<p>Johnfleming: /* Questions */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
===Reading===<br />
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.<br />
<br />
===Doing===<br />
Solve and submit problems <u>1-3</u> and <u>8-10</u> Munkres' book, pages 157-158.<br />
<br />
===Due date===<br />
This assignment is due at the end of class on Monday, October 25, 2010.<br />
<br />
===Suggestions for Good Deeds===<br />
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". <br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
===Questions===<br />
1)Hi, quick question. I am wondering if the term test will cover the material on this assignment, or only the material before the assignment. Thanks! Jason.<br />
<br />
2) In EXAMPLE 7 on page 151 Munkres claims that Rn~ is ['clearly' :)] homeomorphic to Rn:<br />
where Rn~ consists of all sequences x=(x1,x2,x3,...) with xi=0 for i>n,<br />
and Rn consists of all sequences x=(x1,x2,...xn).<br />
Why are they homeomorphic ?? <br />
Thank you kindly. Oliviu.<br />
<br />
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)<br />
<br />
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)<br />
*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<br />
**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.<br />
***Touche, I see your point...that is strange - John<br />
<br />
4)Another question. We know how to show T.S. are homeomorphic. Just find a homeomorphism. But how do we show (0,1) and (0,1] are not homeomorhpic? I d assume they all have induced topology from Rstd. -Kai<br />
<br />
5)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</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Homework_Assignment_410-327/Homework Assignment 42010-10-23T01:05:56Z<p>Johnfleming: /* Questions */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
===Reading===<br />
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.<br />
<br />
===Doing===<br />
Solve and submit problems <u>1-3</u> and <u>8-10</u> Munkres' book, pages 157-158.<br />
<br />
===Due date===<br />
This assignment is due at the end of class on Monday, October 25, 2010.<br />
<br />
===Suggestions for Good Deeds===<br />
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". <br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
===Questions===<br />
1)Hi, quick question. I am wondering if the term test will cover the material on this assignment, or only the material before the assignment. Thanks! Jason.<br />
<br />
2) In EXAMPLE 7 on page 151 Munkres claims that Rn~ is ['clearly' :)] homeomorphic to Rn:<br />
where Rn~ consists of all sequences x=(x1,x2,x3,...) with xi=0 for i>n,<br />
and Rn consists of all sequences x=(x1,x2,...xn).<br />
Why are they homeomorphic ?? <br />
Thank you kindly. Oliviu.<br />
<br />
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)<br />
<br />
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)<br />
*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</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_2110-327/Classnotes for Thursday October 212010-10-22T01:25:37Z<p>Johnfleming: /* Riddle Along */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101021-143325.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]<br />
<br />
==Riddle Along==<br />
Can you color <math>\mathbb{R}^2</math> with 4 colors such that no points with distance one are the same color?<br />
<br />
[[10-327/Solution to coloring R2]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_2110-327/Classnotes for Thursday October 212010-10-22T01:25:07Z<p>Johnfleming: /* Riddle Along */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101021-143325.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]<br />
<br />
==Riddle Along==<br />
Can you color <math>\mathbb{R}^2</math> with 4 colors such that no points with distance one are the same color?<br><br />
[[10-327/Solution to coloring R2]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_2110-327/Classnotes for Thursday October 212010-10-22T01:24:50Z<p>Johnfleming: /* Riddle Along */</p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101021-143325.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]<br />
<br />
==Riddle Along==<br />
Can you color <math>\mathbb{R}^2</math> with 4 colors such that no points with distance one are the same color?<br />
[[10-327/Solution to coloring R2]]</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_2110-327/Classnotes for Thursday October 212010-10-22T01:23:56Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101021-143325.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]<br />
<br />
==Riddle Along==<br />
Can you color <math>\mathbb{R}^2</math> with 4 colors such that no points with distance one are the same color?</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Solution_to_Almost_Disjoint_Subsets10-327/Solution to Almost Disjoint Subsets2010-10-20T00:06:05Z<p>Johnfleming: </p>
<hr />
<div>I think that this collection satisfies the properties.<br />
<br />
Let <math>A</math> be the set of all infinite sequences of 0's and 1's.<br />
Let <math>x \in A</math> and <math> x=x_1,x_2,... </math> with <math> x_i \in \{0,1\}</math><br />
Let <math> p_i </math> be the ith prime number i.e. <math> p_1 = 2, p_2=3, p_3=5, </math> etc.<br />
<br />
Let <math>f:A \to 2^{\mathbb{N}}</math><br />
<br />
Such that <math> f(x)=\bigcup_{n=0}^{\infty}\{\prod_{i=0}^n p_i^{x_i}\}</math><br />
<br />
ie <math>10101010101010.... \to \{2,2*5,2*5*11,2*5*11*17,...\}</math><br />
<br />
Then <math> f(A) </math> is a collection of sets with the desired properties (I think).<br />
<br />
*I don't have time to write out the whole proof, and haven't gone over it completely yet but it seems to work. Showing the the function is injective gives uncountablity. And proving that if they have an infinite intersection they have the same preimage, which is just a single point by injective, they are the same set. - John</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_1410-327/Classnotes for Thursday October 142010-10-19T23:44:26Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101014-142707.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/5/51/10-327-lec09p01.jpg Lecture 9 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/7/76/10-327-lec09p02.jpg Lecture 9 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e3/10-327-lec09p03.jpg Lecture 9 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/9/95/10-327-lec09p04.jpg Lecture 9 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/84/10-327-lec09p05.jpg Lecture 9 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/9/97/10-327-lec09p06.jpg Lecture 9 page 6]<br />
<br />
==Riddles==<br />
===The Dice Game===<br />
Two players A and B decide to play a game. <br />
Player A takes 3 blank dice and labels them with the numbers 1-18.<br />
Player B then picks one of the three die.<br />
Then Player A picks one of the remaining two die.<br />
The players then roll their dice, and the highest number wins the round.<br />
They play 10,023 rounds.<br />
Who would you rather be Player A or B? <br />
<br />
===Almost Disjoint Subsets===<br />
Find an uncountable collection of subsets of <math>\mathbb{N}</math> such that any two subsets only contain a finite number of points in their intersection. Don't cheat by using the axiom of choice! <br />
<br />
[[10-327/Solution to Almost Disjoint Subsets]]<br />
<br />
*I wasn't there for this riddle but it sounded interesting, though I might have the phrasing wrong - John.<br />
* Feel free to cheat and use the axiom of choice - I don't see how it would help anyway. [[User:Drorbn|Drorbn]] 17:40, 18 October 2010 (EDT)</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Class_Photo10-327/Class Photo2010-10-18T21:23:17Z<p>Johnfleming: </p>
<hr />
<div>Our class on September 30, 2010:<br />
<br />
[[Image:10-327-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]<br />
{{10-327/Navigation}}<br />
<br />
Please identify yourself in this photo! There are two ways to do that:<br />
<br />
* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.<br />
* Send [[User:Drorbn|Dror]] an email message with this information.<br />
<br />
The first option is more fun but less private.<br />
<br />
===Who We Are...===<br />
<br />
{| align=center border=1 cellspacing=0<br />
|-<br />
!First name<br />
!Last name<br />
!UserID<br />
!Email<br />
!In the photo<br />
!Comments<br />
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}<br />
<br />
<!--PLEASE KEEP IN ALPHABETICAL ORDER, BY LAST NAME--><br />
<br />
{{Photo Entry|last=Asher|first=Matt|userid=Asher|email=matt.asher utoronto.ca|location=[http://mattasher.com/images/Me_in_photo.jpg Under the arrow] |comments= }}<br />
<br />
{{Photo Entry|last=Darwin|first=Ben|userid=bcd|email=[first.last] @ utoronto.ca|location=Back row, 2nd from left |comments= }}<br />
<br />
{{Photo Entry|last=DiPoce|first=Christian|userid=Cdipoce|email=christian.dipoce@ utoronto.ca|location=very back row, 2nd from right |comments= black t-shirt, yellow writing; as handsome as ever.}}<br />
<br />
{{Photo Entry|last=Dranovski|first=Anne|userid=Anne.d|email=a.dranovski@ utoronto.edu|location=front row, jean jacket. |comments= }}<br />
<br />
{{Photo Entry|last=Fleming|first=John|userid=Johnfleming|email=john.fleming@ utoronto.ca|location=back row seventh from the left. |comments= white collared shirt}}<br />
<br />
{{Photo Entry|last=Hung|first=Ian|userid=Ian|email=ian.hung@ utoronto.ca|location=Right end of second row|comments= I'm wearing a white hoodie}}<br />
<br />
{{Photo Entry|last=Kang|first=Soo Min|userid=soomin_kang|email=soomin.kang@ utoronto.ca|location=2nd row (from front), 2nd (from left)|comments= eyes closed, beige windbreaker}}<br />
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{{Photo Entry|last=Milcak|first=Juraj|userid=milcak|email=j.milcak @ utoronto.edu|location=frontmost, rightmost.|comments= }}<br />
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{{Photo Entry|last=Woodfine|first=Jason|userid=Jdw|email=jason(dot)woodfine (at) utoronto(dot)ca|location=3rd Row (from front), 4th (from left).|comments= Grey collared shirt, arms crossed, glasses.}}<br />
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{{Photo Entry|last=Yang|first=Kai|userid=xwbdsb|email=kai.b.yang@utoronto.ca|location=.|comments= Was planning to take the picture but was late because of washroom...}}<br />
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{{Photo Entry|last=Zhao|first=Frank|userid=Fzhao|email=frank.zhao@ utoronto.ca|location=2nd Row (from front), 6th (from left).|comments= The one in the blue shirt}}<br />
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|}</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_1410-327/Classnotes for Thursday October 142010-10-18T21:11:40Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101014-142707.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/5/51/10-327-lec09p01.jpg Lecture 9 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/7/76/10-327-lec09p02.jpg Lecture 9 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e3/10-327-lec09p03.jpg Lecture 9 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/9/95/10-327-lec09p04.jpg Lecture 9 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/84/10-327-lec09p05.jpg Lecture 9 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/9/97/10-327-lec09p06.jpg Lecture 9 page 6]<br />
<br />
==Riddles==<br />
===The Dice Game===<br />
Two players A and B decide to play a game. <br />
Player A takes 3 blank dice and labels them with the numbers 1-18.<br />
Player B then picks one of the three die.<br />
Then Player A picks one of the remaining two die.<br />
The players then roll their dice, and the highest number wins the round.<br />
They play 10,023 rounds.<br />
Who would you rather be Player A or B? <br />
<br />
===Almost Disjoint Subsets===<br />
Find an uncountable collection of subsets of <math>\mathbb{N}</math> such that any two subsets only contain a finite number of points in their intersection. Don't cheat by using the axiom of choice! <br />
*I wasn't there for this riddle but it sounded interesting, though I might have the phrasing wrong - John.</div>Johnfleminghttp://drorbn.net/index.php?title=10-327/Classnotes_for_Thursday_October_1410-327/Classnotes for Thursday October 142010-10-18T21:10:23Z<p>Johnfleming: </p>
<hr />
<div>{{10-327/Navigation}}<br />
<br />
See some blackboard shots at {{BBS Link|10_327-101014-142707.jpg}}.<br />
<br />
{{Template:10-327:Dror/Students Divider}}<br />
<br />
Here are some lecture notes..<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/5/51/10-327-lec09p01.jpg Lecture 9 page 1]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/7/76/10-327-lec09p02.jpg Lecture 9 page 2]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/e/e3/10-327-lec09p03.jpg Lecture 9 page 3]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/9/95/10-327-lec09p04.jpg Lecture 9 page 4]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/8/84/10-327-lec09p05.jpg Lecture 9 page 5]<br />
<br />
[http://katlas.math.toronto.edu/drorbn/images/9/97/10-327-lec09p06.jpg Lecture 9 page 6]<br />
<br />
==Riddles==<br />
===The Dice Game===<br />
Two players A and B decide to play a game. <br />
Player A takes 3 blank dice and labels them with the numbers 1-18.<br />
Player B then picks one of the three die.<br />
Then Player A picks one of the remaining two die.<br />
The players then roll their dice, and the highest number wins the round.<br />
They play 10,023 rounds.<br />
Who would you rather be Player A or B? <br />
<br />
===Almost Disjoint Subsets===<br />
Find an uncountable collection of subsets of <math>\mathbb{N}</math> such that any two subsets only contain a finite number of points in their intersection. Don't cheat and use the axiom of choice! <br />
*I wasn't there for this riddle but it sounded interesting, though I might have the phrasing wrong - John.</div>Johnfleming