Difference between revisions of "10-327/Homework Assignment 6"
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{{In Preparation}} | {{In Preparation}} | ||
===Reading=== | ===Reading=== | ||
− | '''Read''' sections | + | '''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 | + | Solve and submit the following problems. |
+ | |||
+ | '''Problem 1.''' Problem 1 on page 235 of Munkres' book. | ||
+ | |||
+ | '''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>. | ||
+ | |||
+ | '''Problem 3.''' Show that any function <math>f\colon{\mathbb N}\to I^A</math> from the integers into a "cube" <math>I^A=[0,1]^A</math> has a unique continuous extension to <math>\beta{\mathbb N}</math>. | ||
+ | |||
+ | '''Problem 4.''' Use the fact that there is a countable dense subset within <math>I^I</math> to show that the cardinality of <math>\beta{\mathbb N}</math> is greater than or equal to the cardinality of <math>I^I</math>. | ||
+ | |||
+ | '''Problem 5.''' Show that the cardinality of <math>\beta{\mathbb N}</math> is also less than or equal to the cardinality of <math>I^I</math>, and therefore it is equal to the cardinality of <math>I^I</math>. | ||
+ | |||
+ | '''Problem 6.''' Show that if <math>\mu\in\beta{\mathbb N}\backslash{\mathbb N}</math> and if <math>\mbox{Lim}_\mu</math> is the corresponding generalized limit, and if <math>b</math> is a bounded sequence and <math>f\colon{\mathbb R}\to{\mathbb R}</math> is a continuous function, then <math>\mbox{Lim}_\mu f(b_k) = f(\mbox{Lim}_\mu b_k)</math>. | ||
+ | |||
+ | '''Problem 7.''' Show that there is no super-limit function <math>\mbox{SuperLim}</math> defined on bounded sequences of reals with values in the reals which has the following 3 properties: | ||
+ | # <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>. | ||
+ | # <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>. | ||
+ | # <math>\mbox{SuperLim}(a'_k) = \mbox{SuperLim}(a_k)</math>, where <math>a'</math> is <math>a</math> "shifted once": <math>a'_k=a_{k+1}</math>. | ||
===Due date=== | ===Due date=== |
Revision as of 20:38, 11 November 2010
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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 is homeomorphic to the cantor set .
Problem 3. Show that any function from the integers into a "cube" has a unique continuous extension to .
Problem 4. Use the fact that there is a countable dense subset within to show that the cardinality of is greater than or equal to the cardinality of .
Problem 5. Show that the cardinality of is also less than or equal to the cardinality of , and therefore it is equal to the cardinality of .
Problem 6. Show that if and if is the corresponding generalized limit, and if is a bounded sequence and is a continuous function, then .
Problem 7. Show that there is no super-limit function defined on bounded sequences of reals with values in the reals which has the following 3 properties:
- .
- .
- , where is "shifted once": .
Due date
This assignment is due at the end of class on Thursday, November 18, 2010.
Dror's notes above / Student's notes below |