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 4 properties:

1. ${\mbox{SuperLim}}(a_{k})=\lim a_{k}$, if the sequence $a_{k}$ is convergent.
2. ${\mbox{SuperLim}}(a_{k}+b_{k})={\mbox{SuperLim}}(a_{k})+{\mbox{SuperLim}}(b_{k})$.
3. ${\mbox{SuperLim}}(a_{k}\cdot b_{k})={\mbox{SuperLim}}(a_{k})\cdot {\mbox{SuperLim}}(b_{k})$.
4. ${\mbox{SuperLim}}(a'_{k})={\mbox{SuperLim}}(a_{k})$, 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)

• One way to do this is that if you have a map $f(x):A\to B$ such that f(x) is 1-1, then $A$ has cardinality less than or equal to $B$. I haven't worked on this question yet, so it may be a red herring, but in general I find the technique useful. John

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)

• I actually genuinely enjoyed writing this assignment, thinking that I've asked a lovely collection of questions that will challenge you in the positive sense of the word - make you scratch your heads, and struggle a bit, and fight a bit to understand what is going on and eventually get some satisfaction and gain some appreciation of the intricacy and beauty of the subject matter as you understand and solve these problems. So please challenge yourself and think and struggle a bit more. Drorbn 10:32, 13 November 2010 (EST)

• I have a question concerning Problem 7. Perhaps I am misunderstanding it, but if we just define $SuperLim(a_{k})=0$ for all sequences the 3 properties seem to hold, maybe that $SuperLim(a_{k})=Lim(a_{k})$ if the sequence does converge should be added? Or maybe just ruling out constant functions is suffient... -John
  • Thanks! You are absolutely right. The question has been modified. Drorbn 15:23, 13 November 2010 (EST)