Difference between revisions of "10-327/Homework Assignment 5"

From Drorbn
Jump to: navigation, search
(Solution)
(EDIT: moved Kai's HW5 solutions to new page)
 
Line 13: Line 13:
 
*Question. In 4. By bounded metric space you mean there exists a point and an epsilon where this epsilon nbd contains everything in the metric space? -Kai
 
*Question. In 4. By bounded metric space you mean there exists a point and an epsilon where this epsilon nbd contains everything in the metric space? -Kai
 
** Indeed so, though usually when talking about boundedness, people use the letter <math>M</math> and not the letter <math>\epsilon</math>. It makes no difference, of course.
 
** Indeed so, though usually when talking about boundedness, people use the letter <math>M</math> and not the letter <math>\epsilon</math>. It makes no difference, of course.
 
===Solution===
 
[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a501.JPG page1]
 
[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a502.JPG page2]
 
[http://katlas.math.toronto.edu/drorbn/images/6/6b/10-327a503.JPG page3]
 
[http://katlas.math.toronto.edu/drorbn/images/6/61/10-327a504.JPG page4]
 
[http://katlas.math.toronto.edu/drorbn/images/1/1d/10-327a505.JPG page5]
 
[http://katlas.math.toronto.edu/drorbn/images/c/c6/10-327a506.JPG page6]
 
[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a507.JPG page7]
 
[http://katlas.math.toronto.edu/drorbn/images/d/df/10-327a508.JPG page8]
 
[http://katlas.math.toronto.edu/drorbn/images/9/9d/10-327a509.JPG page9]
 
[http://katlas.math.toronto.edu/drorbn/images/8/8e/10-327a510.JPG page10]
 
[http://katlas.math.toronto.edu/drorbn/images/7/75/10-327a511.JPG page11]
 
[http://katlas.math.toronto.edu/drorbn/images/7/70/10-327a512.JPG page12]
 
[http://katlas.math.toronto.edu/drorbn/images/d/dc/10-327a513.JPG page13]
 
[http://katlas.math.toronto.edu/drorbn/images/4/4d/10-327a514.JPG page14]
 
[http://katlas.math.toronto.edu/drorbn/images/c/c5/10-327a515.JPG page15]
 
[http://katlas.math.toronto.edu/drorbn/images/6/6d/10-327a516.JPG page16]
 
[http://katlas.math.toronto.edu/drorbn/images/c/c6/10-327a517.JPG page17]
 
[http://katlas.math.toronto.edu/drorbn/images/d/d5/10-327a518.JPG page18]
 
 
An assignment without a solution is like a nightmare to me. I like every question accompanied
 
with a clean solution aside.(Might not be the case for research question because they are just
 
simply too hard.) I would like to share this happiness of understanding and acquiring knowledge
 
with everybody because I don't think this class is a battle. I certainly don't like the idea that
 
we should keep information/answers as something like business secrets. This learning process
 
should be enjoyable which should be full of discussions instead of things like "you have to think on
 
your own/ I can't tell you the answer". I know maybe other people might not agree with me but I believe
 
a positive learning environment is crucial to truly understanding something well although we should not neglect
 
independent thinking at the same time. That is why I share whatever I have with you. If I am wrong feel free to criticize
 
me, and I am pretty sure a lot of people don't agree with me. But that is OK because there is just no
 
absolute right or wrong and everybody is doing what they think is right. Just like you can't say if
 
Axiom of choice is right or not. If you believe then it is right. If you don't believe it then it is wrong.
 
Or maybe, for the entire subject--math, if you believe it then it is right. If you don't believe it then it
 
is completely wrong.-Kai
 

Latest revision as of 22:54, 10 December 2010

Reading

Read sections 26 and 27 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 28 and 29, just to get a feel for the future.

Doing

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

Due date

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

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