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 ${\mathbb {R} }$, HW4, HW4 Solutions, Thursday - connectedness, path-connectedness and products 7 Oct 25 Monday - Compactness of $[0,1]$, 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 - $T_{3.5}$ and $I^{A}$, 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 $\{0,1\}^{\mathbb {N} }$ 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)