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\title{Math 112 Problem Set 6}
\author{Due {\bf Friday} March 9, 2018}

\maketitle

{\bf Core problems:}
\begin{enumerate} 
	\item Find an open cover of the following with no finite subcover
	\begin{enumerate}
		\item $\{\vec{x} \in \R^n \mid \|\vec{x}\| < 1\}$
		\item $\Z$ in $\R$
	\end{enumerate}(3, \#5)
	\item We generalize a previous problem to $\R^n$: Let $\{F_k\}_k$ be a sequence of compact sets in $\R^n$ such that
	\begin{enumerate}
		\item $diam(F_k) \to 0$
		\item $F_{k+1} \subset F_k$ for $k \in \N$
	\end{enumerate}
	Show that there is exactly one point in $\cap_{k=1}^\infty F_k$.  (3, \#6; $diam(A) = \sup\{\|\vec{x}-\vec{y}\| \mid x, y\in A\}$)
	\item Prove or find a counter-example to each of the following for $A \subset \R^n$:
	\begin{enumerate}
		\item $A$ is compact implies $\R^n-A$ is connected
		\item $A$ is connected implies $\R^n-A$ is connected
		\item $A$ is connected implies $\R^n-A$ is open or closed
		\item $\R^n-\{\vec{x} \mid \|x\|\leq 1\}$ is connected
	\end{enumerate}(3, \# 9)
	\item Let $A \subset \R^n$ and $B \subset \R^m$ be compact.  Show that $A \times B \subset \R^{n+m}$ is compact. (3, \# 15)
	\item Let $A \subset \R^n$ and $B \subset \R^m$ be path-connected.  Show that $A \times B \subset \R^{n+m}$ is path-connected. (3, \# 15)
	\item $A \subset \R^n$ is called \emph{nowhere dense} iff \underline{$int(cl(A))=\emptyset$}.  Show that there does {\bf not} exist a sequence $\{A_n\}_n$ of nowhere dense subsets of $\R^n$ such that $\R^n = \cup_{n=1}^\infty A_n$.(3, \#33)
	\item Let $A, B \subset \R^n$ with $A$ compact and $B$ closed such that $A \cap B = \emptyset$.  We previously defined the distance between a point and a set and now define the distance between these sets as
	$$dist(A, B) = \inf \{d(x, y) \mid x \in A, y \in B\}$$
	\begin{enumerate}
		\item Show that $dist(A, B) \neq 0$.
		\item Can you find closed $A, B$ that are disjoint but the distance between them is zero?
	\end{enumerate}(3, \#37)
	\item Recall the definition of the Cantor set $C$ from class (or see the book).  Prove the following things about it:
	\begin{enumerate}
		\item $C$ is compact
		\item $C$ has infinitely many points
		\item $int(C) = \emptyset$
		\item $C$ is closed and has no isolated points (this is called being \emph{perfect})
		\item for any \underline{$x \neq y \in C$, there are $U, V \subset C$ that are disjoint and open such that $x \in U$} \underline{and $y\in V$ and $C \subset U \cup V$} (this is called being \emph{totally disconnected})
	\end{enumerate}(3, \#39)
\end{enumerate}

{\bf Niche problems:}
\begin{enumerate}
	\item Define the `Topologist's sine curve' to be 
	$$\mathbb{T} = \{(0,y) \mid y \in [-1, 1]\} \cup \left\{\left(x, \sin \frac{1}{x}\right) \mid x > 0 \right\} \subset \R^2$$
	In class, we claimed that this is connected but not path-connected.  Prove this.
	
	(You can use your knowledge about $\sin$ for this problem.)
	
	\item We expand the above discussion of the Cantor set $C$.  We take for granted that every number in $[0,1]$ has a representation in base 3 (and you can use facts about bases in this problem).  That is, given $x \in [0, 1]$, there is some sequence $\{b_n\}_n$ whose terms consist of $0, 1,$ or $2$ such that $x = \sum_{n=1}^\infty \frac{b_n}{3^n}$ (although this sequence is not unique).  Show that $x \in C$ iff it has a base 3 representation consisting only of $0$'s and $2$'s.
	
	

	\item Lest you think we were done, here is an alternate way of constructing $\R$ from $\Q$ using Cauchy sequences.  This method has the advantage that it's actually a general method of taking a metric space $(X, d)$ and enlarging it to a complete metric space $(\bar{X}, \bar{d})$.  Recall the definition of an equivalence relation from last time.\\
	
	Let $R$ be an equivalence relation on a set $A$.  Given $a \in A$, the equivalence class of $a$ is the subset of $A$ consisting of everything that is $R$-equivalent to $a$:
	$$[a]_R = \{b \in A \mid (a, b) \in R\}$$
	Then we can form the set of \emph{equivalence classes} 
	$$A/R = \left\{ [a]_R \mid a \in A\right\}$$
	which has equivalence classes as its elements.\\
	
	Show that $A/R$ is a partition of $A$.  That is, every point in $a$ is in one of the sets and that different classes are disjoint.
	
	\item Let $\cC$ be the set of Cauchy sequence with elements in $\Q$.  Define an equivalence relation $\sim$ on $\cC$ as follows: given Cauchy sequences $\{a_n\}_n, \{b_n\}_n \subset \Q$, we set
	$$\{a_n\}_n \sim \{b_n\}_n \text{ iff } \text{ for every rational $\epsilon > 0$, there is $N$ such that if $n > N$, we have $|a_n - b_n|<\epsilon$}$$
	This is an equivalence relation (you don't have to show this, it's the equivalence relation from the previous assignment written to avoid $\R$).
	
	Similar to Dedekind cuts, we wish to turn $\cC/\sim$ into a complete ordered field.  Define $\oplus, \otimes, \triangleleft$ on equivalence classes of Cauchy sequences as follows:
	\begin{eqnarray*}
	[\{a_n\}]_{\sim} \oplus [\{b_n\}]_\sim &=& [\{a_n + b_n\}]_\sim \\
	{[}\{a_n\}{]}_\sim \otimes [\{b_n\}]_\sim &=& [\{a_n b_n\}]_\sim\\
	{[}\{a_n\}]_\sim \triangleleft [\{b_n\}]_\sim &\text{ iff }& \text{ there is $\epsilon > 0$ and $N$ such that $n > N$ implies $a_n + \epsilon < b_n$}
	\end{eqnarray*}
	But there's a (potential) problem! We're defining functions on equivalence classes of Cauchy sequences, but using the particular sequence that we've picked as a `representative' to define the function.  This is a problem if equivalent Cauchy sequences sum to inequivalent sequences.  Show that this can't happen and that the functions are \emph{well-defined}.
	
	In particular, let $\{a_n\} \sim \{a'_n\}$ and $\{b_n\} \sim \{b'_n\}$ be Cauchy sequences.  Show that
	\begin{enumerate}
		\item $\{a_n + b_n\} \sim \{a'_n +b'_n\}$
		\item $\{a_n b_n\} \sim \{a'_n b'_n\}$
		\item $[\{a_n\}]_\sim \triangleleft [\{b_n\}] \iff [\{a'_n\}]_\sim \triangleleft [\{b'_n\}]_\sim$
	\end{enumerate}
	
\end{enumerate}

{\bf Doc Brown problems:}
\begin{enumerate}
	\item Have a good Spring Break! (this is worth no class points, but lots of life points)
\end{enumerate}

For problems from the book, something like 1, \#8 refers to \#8 from the exercises at the end of Chapter 1, while something like 1.3, \#1 refers to \#1 from the exercises at the end of section 1.3.
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