Drorbn - User contributions [en]

10-327/Homework Assignment 4

2010-10-23T14:00:33Z

Fzhao: /* Questions */

{{10-327/Navigation}}

===Reading===
Read sections 23 through 25 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 26 through 27, just to get a feel for the future.

===Doing===
Solve and submit problems 1-3 and 8-10 Munkres' book, pages 157-158.

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

===Suggestions for Good Deeds===
Annotate our Monday videos (starting with {{10-327/vp|0927}}) in a manner similar to (say) {{dbnvp link|AKT-090910-1|AKT-090910-1}}, and/or add links to the blackboard shots, in a manner similar to {{dbnvp link|Alekseev-1006-1|Alekseev-1006-1}}. Also, make ''constructive'' suggestions to me, {{Dror}} and / or the videographer, Qian (Sindy) Li, on how to improve the videos and / or the software used to display them. Note that "constructive" means also, "something that can be implemented relatively easily in the real world, given limited resources".

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

===Questions===
1)Hi, quick question. I am wondering if the term test will cover the material on this assignment, or only the material before the assignment. Thanks! Jason.

2) In EXAMPLE 7 on page 151 Munkres claims that Rn~ is ['clearly' :)] homeomorphic to Rn:
where Rn~ consists of all sequences x=(x1,x2,x3,...) with xi=0 for i>n,
and Rn consists of all sequences x=(x1,x2,...xn).
Why are they homeomorphic ??
Thank you kindly. Oliviu.

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)

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

4)Question about the proof for [0,1] being connected. A few details are omitted. why would a closed subset of [0,1] contain its supremum? Also why [0,g_0] being a subset of A follows automatically after we showed that g_0 is in A? -Kai
*1. Suppose <math>S </math> is closed in <math> [0,1]. \Rightarrow S^C </math> is open. If <math> sup(S)=\alpha \notin S \Rightarrow \exists r>0 </math> s.t. <math> B(\alpha, r) \subset S^C \Rightarrow \alpha - 0.5r \in S^C \Rightarrow \alpha - 0.5r < \alpha </math> is an upper bound for S. <math> \Rightarrow \Leftarrow </math>
*2. Recall that <math> G = \{g | [0,g] \subset A\}; g_0 = sup(G) \Rightarrow \forall g < g_0, [0,g] \subset A \Rightarrow [0, g_0) \subset A </math>. So, if <math> g_0 \in A \Rightarrow [0,g_0] \subset A</math>. -Frank [[User:Fzhao|Fzhao]] 23:50, 22 October 2010 (EDT)

**Thanks Frank. But I don't think your solution is convincing enough. \alpha - 0.5r is indeed not in S but why can you say it is an upper bound for S? Remember S could be rather complicated set all you know is that it is closed.
for 2 why is \Rightarrow \forall g < g_0, [0,g]? even if g_0 is sup(G) that does not mean anything less than g_0 would be in G. Consider [0,1] union {3}.

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

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

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

10-327/Homework Assignment 4

2010-10-23T13:58:13Z

Fzhao: /* Questions */

{{10-327/Navigation}}

===Reading===
Read sections 23 through 25 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 26 through 27, just to get a feel for the future.

===Doing===
Solve and submit problems 1-3 and 8-10 Munkres' book, pages 157-158.

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

===Suggestions for Good Deeds===
Annotate our Monday videos (starting with {{10-327/vp|0927}}) in a manner similar to (say) {{dbnvp link|AKT-090910-1|AKT-090910-1}}, and/or add links to the blackboard shots, in a manner similar to {{dbnvp link|Alekseev-1006-1|Alekseev-1006-1}}. Also, make ''constructive'' suggestions to me, {{Dror}} and / or the videographer, Qian (Sindy) Li, on how to improve the videos and / or the software used to display them. Note that "constructive" means also, "something that can be implemented relatively easily in the real world, given limited resources".

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

===Questions===
1)Hi, quick question. I am wondering if the term test will cover the material on this assignment, or only the material before the assignment. Thanks! Jason.

2) In EXAMPLE 7 on page 151 Munkres claims that Rn~ is ['clearly' :)] homeomorphic to Rn:
where Rn~ consists of all sequences x=(x1,x2,x3,...) with xi=0 for i>n,
and Rn consists of all sequences x=(x1,x2,...xn).
Why are they homeomorphic ??
Thank you kindly. Oliviu.

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)

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

4)Question about the proof for [0,1] being connected. A few details are omitted. why would a closed subset of [0,1] contain its supremum? Also why [0,g_0] being a subset of A follows automatically after we showed that g_0 is in A? -Kai
*1. Suppose <math>S </math> is closed in <math> [0,1]. \Rightarrow S^C </math> is open. If <math> sup(S)=\alpha \notin S \Rightarrow \exists r>0 </math> s.t. <math> B(\alpha, r) \subset S^C \Rightarrow \alpha - 0.5r \in S^C \Rightarrow \alpha - 0.5r < \alpha </math> is an upper bound for S. <math> \Rightarrow \Leftarrow </math>
*2. Recall that <math> G = \{g | [0,g] \subset A\}; g_0 = sup(G) \Rightarrow \forall g < g_0, [0,g] \subset A \Rightarrow [0, g_0) \subset A </math>. So, if <math> g_0 \in A \Rightarrow [0,g_0] \subset A</math>. -Frank [[User:Fzhao|Fzhao]] 23:50, 22 October 2010 (EDT)

**Thanks Frank. But I don't think your solution is convincing enough. \alpha - 0.5r is indeed not in S but why can you say it is an upper bound for S? Remember S could be rather complicated set all you know is that it is closed.
for 2 why is \Rightarrow \forall g < g_0, [0,g]? even if g_0 is sup(G) that does not mean anything less than g_0 would be in G. Consider [0,1] union {3}.

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

*For the second question, notice that the supremum is the least upper bound (of G), so <math> \forall r > 0, \exists g \in G \cap B(g_0,r) \Rightarrow [0,g] \subset A \Rightarrow [0,g'] \subset A, \forall g' < g \Rightarrow g' \in G \Rightarrow \forall g < g_0 </math>, take <math> r = 0.5(g_0 - g) \Rightarrow \exists a \in (g_0 - r, g_0) \cap G, \Rightarrow [0,g] \subset [0,a] \subset A.</math>

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

10-327/Homework Assignment 4

2010-10-23T03:50:40Z

Fzhao: /* Questions */

10-327/Homework Assignment 4

2010-10-23T03:39:33Z

Fzhao: /* Questions */

10-327/Class Photo

2010-10-02T19:27:27Z

Fzhao: /* Who We Are... */

Our class on September 30, 2010:

[[Image:10-327-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{10-327/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

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

The first option is more fun but less private.

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

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



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

{{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}}
|}

10-327/Class Photo

2010-10-02T19:27:05Z

Fzhao: /* Who We Are... */

Our class on September 30, 2010:

[[Image:10-327-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{10-327/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

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

The first option is more fun but less private.

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

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



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

{{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}}
|}

10-327/Homework Assignment 2

2010-10-01T03:42:42Z

Fzhao:

{{10-327/Navigation}}

===Reading===
Read sections 17 through 21 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 22 through 24, just to get a feel for the future.

===Doing===
Solve the following problems from Munkres' book, though submit only the underlined ones: Problems 6, 7, 8, 13, 14, 19abc, 19d, 21 on pages 101-102, and problems 7a, 7b, 8, 9ab, 9c, 13 on pages 111-112.

===Due date===
This assignment is due at the end of class on Thursday, October 5, 2010.

===Suggestions for Good Deeds===
Annotate our Monday videos (starting with {{10-327/vp|0927}}) in a manner similar to (say) {{dbnvp link|AKT-090910-1|AKT-090910-1}}, and/or add links to the blackboard shots, in a manner similar to {{dbnvp link|Alekseev-1006-1|Alekseev-1006-1}}. Also, make ''constructive'' suggestions to me, {{Dror}} and / or the videographer, Qian (Sindy) Li, on how to improve the videos and / or the software used to display them. Note that "constructive" means also, "something that can be implemented relatively easily in the real worlds, given limited resources".

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

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