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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 1} \vs \\
{\bf due Wednesday 9/11/2013} \vs \vs
\end{center}

\noindent
Unless otherwise stated, $X$ is a metric space with a given distance function $d$. \vs \vs

\noindent
{\bf Problem 1.} For $r>0$, define the $r$-{\bit neighborhood} of $E \subset X$ by
$$
N_r(E)=\{ p \in X: d(p,q)<r \ \text{for some} \ q \in E \}. 
$$
\begin{enumerate}
\item[(i)]
Show that $N_r(E)$ is open. \vs
\item[(ii)]
Show that $\ov{E} \subset N_r(E)$. \vs
\item[(iii)]
What is $\bigcap_{r>0} N_r(E)$? \vs 
\end{enumerate}

\noindent
{\bf Problem 2.} 
\begin{enumerate}
\item[(i)]
Take three sets $G \subset E \subset F$ in $X$, with $G$ open and $F$ closed. Show that 
$$
G \subset \mathring{E} \subset \ov{E} \subset F. 
$$
\item[(ii)]
Prove that for any collection $\{ E_{\alpha} \}$ of sets in $X$, 
$$
\bigcup_{\alpha} \ov{E_{\alpha}} \subset \ov{\bigcup_{\alpha} E_{\alpha}} \qquad \text{and} \qquad \bigcap_{\alpha} \ov{E_{\alpha}} \supset \ov{\bigcap_{\alpha} E_{\alpha}}.
$$
Show by examples that both inclusions are proper.\vs  
\item[(iii)]
Prove that for any finite collection $\{ E_1, \ldots, E_n \}$ of sets in $X$,
$$
\bigcup_{1 \leq i \leq n} \ov{E_i} = \ov{\bigcup_{1 \leq i \leq n} E_i}. \vs
$$
\end{enumerate}

\noindent
{\bf Problem 3.} 
\begin{enumerate}
\item[(i)]
The {\bit boundary} of $E$, denoted by $\bd E$, is the set of all $p \in X$ for which every neighborhood meets both $E$ and its complement $E^c$. Verify the relations $\bd E = \ov{E} \cap \ov{E^c}$ and $\ov{E}=\mathring{E} \cup \bd E$; in particular $\bd E$ is closed. \vs 
\item[(ii)]
Can the unit interval $[0,1]$ in $\RR$ be the boundary of a set? What if one replaces $[0,1]$ with other closed sets in $\RR$? \vs
\end{enumerate}

\noindent
{\bf Problem 4.} Let $(X,d)$ and $(\hat{X},\hat{d})$ be metric spaces. Verify that each of the following defines a metric on the product $X \times \hat{X}$:
\begin{align*}
d_{\infty}((p,\hat{p}), (q,\hat{q})) & = \max \{ d(p,q), \hat{d}(\hat{p},\hat{q}) \} \\
d_1((p,\hat{p}), (q,\hat{q})) & = d(p,q)+\hat{d}(\hat{p},\hat{q}) \\
d_2((p,\hat{p}), (q,\hat{q})) & = (d(p,q)^2+\hat{d}(\hat{p},\hat{q})^2)^{1/2} 
\end{align*}
Prove the inequalities
$$
d_{\infty} \leq d_2 \leq d_1 \leq 2 d_{\infty}.  
$$
Conclude that these metrics define the same open sets on $X \times \hat{X}$. \vs

\noindent
{\bf Problem 5.} 
\begin{enumerate}
\item[(i)]
Suppose $\{ p_n \}$ and $\{ q_n \}$ are sequences in $X$ such that $p_n \to p$ and $q_n \to q$. Show that $d(p_n,q_n) \to d(p,q)$. \vs
\item[(ii)]
Show that a Cauchy sequence with a convergent subsequence is convergent. \vs
\end{enumerate}       

\noindent
{\bf Problem 6.} Let $\{ p_n \}$ be a sequence in a complete metric space $X$. Suppose there are constants $C>0$ and $0<\lambda<1$ such that $d(p_n,p_{n+1})< C \lambda^n$ for all $n$. Show that $\{ p_n \}$ converges in $X$.  






\end{document}
