Difference between revisions of "10-327/Homework Assignment 6"

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{{In Preparation}}
 
{{In Preparation}}
 
===Reading===
 
===Reading===
'''Read''' sections ??? 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.
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'''Read''' sections 37-38 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 30-33, just to get a feel for the future.
  
 
===Doing===
 
===Doing===
Solve the following problems from Munkres' book, though submit only the <u>underlined</u> ones: Problems ??? on pages ???.
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Solve and submit the following problems.
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'''Problem 1.''' Problem 1 on page 235 of Munkres' book.
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'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.
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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>.
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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>.
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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>.
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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>.
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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 3 properties:
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# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.
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# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.
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# <math>\mbox{SuperLim}(a'_k) = \mbox{SuperLim}(a_k)</math>, where <math>a'</math> is <math>a</math> "shifted once": <math>a'_k=a_{k+1}</math>.
  
 
===Due date===
 
===Due date===

Revision as of 20:38, 11 November 2010

In Preparation

The information below is preliminary and cannot be trusted! (v)

Reading

Read sections 37-38 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 30-33, just to get a feel for the future.

Doing

Solve and submit the following problems.

Problem 1. Problem 1 on page 235 of Munkres' book.

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

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

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

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

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

Problem 7. Show that there is no super-limit function \mbox{SuperLim} defined on bounded sequences of reals with values in the reals which has the following 3 properties:

  1. \mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k).
  2. \mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k).
  3. \mbox{SuperLim}(a'_k) = \mbox{SuperLim}(a_k), where a' is a "shifted once": a'_k=a_{k+1}.

Due date

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

Dror's notes above / Student's notes below