10-327/Homework Assignment 6

DBN: Classes: 2010-11: 10-327/Navigation Additions to the MAT 327 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 13 About This Class, Monday - Continuity and open sets, Thursday - topologies, continuity, bases. 2 Sep 20 Monday - More on bases, Thursdsay - Products, Subspaces, Closed sets, HW1, HW1 Solutions 3 Sep 27 Monday - the Cantor set, closures, Thursday, Class Photo, HW2, HW2 Solutions 4 Oct 4 Monday - the axiom of choice and infinite product spaces, Thursday - the box and the product topologies, metric spaces, HW3, HW3 Solutions 5 Oct 11 Monday is Thanksgiving. Thursday - metric spaces, sequencial closures, various products. Final exam's date announced on Friday. 6 Oct 18 Monday - connectedness in [math]\displaystyle{ {\mathbb R} }[/math], HW4, HW4 Solutions, Thursday - connectedness, path-connectedness and products 7 Oct 25 Monday - Compactness of [math]\displaystyle{ [0,1] }[/math], Term Test on Thursday, TT Solutions 8 Nov 1 Monday - compact is closed and bounded, maximal values, HW5, HW5 Solutions, Wednesday was the last date to drop this course, Thursday - compactness of products and in metric spaces, the FIP 9 Nov 8 Monday-Tuesday is Fall Break, Thursday - Tychonoff and a taste of Stone-Cech, HW6, HW6 Solutions 10 Nov 15 Monday - generalized limits, Thursday - Normal spaces and Urysohn's lemma, HW7, HW7 Solutions 11 Nov 22 Monday - [math]\displaystyle{ T_{3.5} }[/math] and [math]\displaystyle{ I^A }[/math], Thursday - Tietze's theorem 12 Nov 29 Monday - compactness in metric spaces, HW8, HW8 Solutions, Thursday - completeness and compactness 13 Dec 6 Monday - Baire spaces and no-where differentiable functions, Wednesday - Hilbert's 13th problem; also see December 2010 Schedule R Dec 13 See December 2010 Schedule F Dec 20 Final exam, Monday December 20, 2PM-5PM, at BR200 Register of Good Deeds Add your name / see who's in! See Hilbert's 13th

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 [math]\displaystyle{ \{0,1\}^{\mathbb N} }[/math] is homeomorphic to the cantor set [math]\displaystyle{ C }[/math].

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

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

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

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

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

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

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)