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\title{Math 3110: Homework 6}
%\author{Zach Norwood}
\date{Due \textcolor{navy}{Wednesday, 13 March 2019}}

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\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.


\begin{problem}
 Recall the version of the Heine--Borel Theorem we proved in lecture:
 
 \begin{thm*}[Heine--Borel]
  Suppose that to each $x\in [a,b]$ is assigned a positive ``jurisdiction radius'' $\delta(x)>0$.
  (That is, $\delta$ is a function $[a,b]\to (0,\infty)$.)
  There are $n\in\N$ and finitely many $x_1,\dots,x_n\in[a,b]$ such that
  \begin{equation}	\label{hb}		\tag{\textsc{hb}}
   [a,b] \subs B_{\delta(x_1)}(x_1) \cup \cdots \cup B_{\delta(x_n)}(x_n).
  \end{equation}
 \end{thm*}

 \begin{enumerate} [(a)]
  \item When we prove something of the form ``for all $x$ there is a $y$\dots'' it is usually the case
  that the $y$ will depend on $x$. 
  Show, by giving a family of examples, that the number $n$ in the conclusion of the Heine--Borel Theorem
  depends on $\delta$. More specifically, find for each $n\ge1$ a function $\delta_n\colon[0,1]\to(0,\infty)$
  such that at least $n$ (and no fewer) elements of $[0,1]$ are needed to satisfy~\eqref{hb}.
  
  \item Show that the Heine--Borel Theorem does not apply to the open interval $(0,1)$.
  That is, construct a function $\delta\colon(0,1)\to(0,\infty)$ such that no finitely many elements
  $x_1,\dots,x_n$
  can be chosen from $(0,1)$ to satisfy~\eqref{hb}.
  
  \item Show that the Heine--Borel Theorem does not apply to the set $[1,2]\cap\Q$.
  That is, construct a function $\delta\colon[1,2]\cap\Q\to(0,\infty)$ such that no finitely many elements
  $x_1,\dots,x_n$
  can be chosen from $[1,2]\cap\Q$ to satisfy~\eqref{hb}.
 \end{enumerate}
\end{problem}

\begin{problem}
 Suppose that $X$ is a bounded infinite set. Prove that $X$ has at least one cluster point.
 (\emph{Hint}: Use your theorems.)
\end{problem}

\begin{problem}
 Show that there are \emph{not} two nonempty disjoint open subsets of $\R$ whose union is all of $\R$.
\end{problem}

\begin{problem}
 Show that no set has uncountably many isolated points.
 
 (\emph{Hint}: Homework 3 \#4(c) might be helpful.)
\end{problem}

\begin{problem}
\hfill
 \begin{enumerate} [(a)]
 \item
 Recall from lecture that the Cantor set $\mathcal{C}$ was defined to be the intersection
 of sets $C_n$, where $C_n$ is a disjoint union of $2^n$ intervals, each of diameter $\tfrac{1}{3^n}$.
 For each $n\in\N$ let $E_n$ be the set of ($2^{n+1}$ many) endpoints of the intervals 
 that comprise $C_n$. (E.g.\ $E_2 = \{0,\tfrac19,\tfrac29,\tfrac13,\tfrac23,\tfrac79,\tfrac89,1\}$.)
 Notice that $\bigcup_{n\in\N} E_n$ is a countable subset of $\mathcal{C}$.\footnote{%
 This follows from facts from lecture. You should prove it, but \emph{don't} turn in your proof with your
 homework.} 
 Show that the closure of $\bigcup_{n\in\N} E_n$ is $\mathcal{C}$.
 
 \item
 Let $\mathcal{C}$ be the Cantor set. Show that $\mathcal{C}$ is exactly the set of 
 numbers in $[0,1]$ whose ternary (i.e., base-$3$) expansion uses only $0$s and $2$s.
 That is, show that $x\in\mathcal{C}$ iff there is a sequence
 $(a_n)_{n\in\N}$ with $a_n\in\{0,2\}$ for all $n$, such that 
 $x = \sum_{n\ge0} \frac{a_n}{3^n}$.

 \item
 Let $\mathcal{C}$ be the Cantor set.
 Prove (perhaps by using the previous part or by following the hints given in Exercise 3.3.7
 in the textbook) that $\mathcal{C}+\mathcal{C} = [0,2]$.
 (By $\mathcal{C}+\mathcal{C}$, I mean the set $\set{ x+y }{ x,y\in\mathcal{C}}$.)
 \end{enumerate}
\end{problem}

\end{document}




































