10-327/Homework Assignment 6

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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

Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that? 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? Xwbdsb 00:25, 13 November 2010 (EST)

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??Xwbdsb 00:36, 13 November 2010 (EST) I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is not the standard definition in our topology class? Sorry Dror I am sort of lost of what to do for this assignment...Xwbdsb 00:36, 13 November 2010 (EST)

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? Xwbdsb 00:58, 13 November 2010 (EST)