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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 2} \vs \\
{\bf due Friday 9/20/2013} \vs \vs
\end{center}

\noindent
{\bf Problem 1.} Give a direct proof for compactness of $[a,b] \subset \RR$ by completing the following sketch: Let $\{ U_{\alpha} \}$ be an open cover for $[a,b]$ and define
$$
E= \{ x \in [a,b]: [a,x] \ \text{is covered by finitely many of the} \ U_\alpha \}.
$$
Show that $b \in E$ by proving $E$ is closed and $\sup E =b$. \vs

\noindent
{\bf Problem 2.} Suppose $K \subset U \subset X$, with $K$ compact and $U$ open. Show that $N_\ve(K) \subset U$ if $\ve>0$ is sufficiently small. Here $N_\ve(K)$ is the $\ve$-neighborhood of $K$ defined by $N_\ve(K) = \bigcup_{p \in K} B(p,\ve)$. \vs

\noindent
{\bf Problem 3.} The previous result has a useful generalization in which a single open set is replaced by an arbitrary open cover. Suppose $K$ is compact and $\{ U_{\alpha} \}$ is an open cover for $K$. Show that there is an $\ve>0$ such that for every $p \in K$, the ball $B(p,\ve)$ is contained in some $U_{\alpha}$. Such $\ve$ is called a {\bit Lebesgue number} for the cover $\{ U_{\alpha} \}$. \vs 

Give an example of an open cover for a non-compact set in $\RR$ which does not have a Lebesgue number. \vs

\noindent
{\bf Problem 4.} 
\begin{enumerate}
\item[(i)]
Prove the following version of Cantor's theorem: {\it If $X$ is a complete metric space, for every sequence $E_1 \supset E_2 \supset E_3 \supset \cdots$ of non-empty closed sets in $X$ with  $\diam E_n \to 0$, the intersection $\bigcap_{n \geq 1} E_n$ is a single point.} \vs

\item[(ii)]
Show by an example that $\bigcap_{n \geq 1} E_n$ may be empty if the condition $\diam E_n \to 0$ is dropped. \vs

\item[(iii)]
Show that the property in (i) characterizes complete metric spaces. That is, prove that if every decreasing sequence of non-empty closed sets with shrinking diameters nests down to a point, then the ambient space must be complete. \vs 
\end{enumerate}
(Hint: For (i), take a point $p_n \in E_n$ and show that $\{ p_n \}$ is a Cauchy sequence. For (iii), take a Cauchy sequence $\{ p_n \}$ in $X$ and let $E_n$ be the closure of $\{ p_n, p_{n+1}, p_{n+2}, \ldots \}$.) \vs 

\noindent
{\bf Problem 5.} A metric $d$ on a set $X$ is called an {\bit ultra-metric} if it satisfies the stronger form of the triangle inequality
$$
d(x,y) \leq \max \{ d(x,z) , d(z,y) \} \qquad \text{for all} \ x,y,z \in X. 
$$
Suppose $d$ is an ultra-metric on $X$. \vs
\begin{enumerate}
\item[(i)]
Show that if $d(x,z) \neq d(z,y)$, then $d(x,y) = \max \{ d(x,z) , d(z,y) \}$. \vs
\item[(ii)]
Show that a sequence $\{ p_n \}$ in $(X,d)$ is Cauchy iff $d(p_n,p_{n+1}) \to 0$ as $n \to \infty$. \vs
\end{enumerate} 

\noindent
{\bf Problem 6.} Let $X$ be the space of all sequences $P = \{ p_n \}_{n \geq 1}$ in a given set. For $P=\{ p_n \}$ and $Q= \{ q_n \}$ in $X$, let $n(P,Q)$ be the smallest index $n \in \NN$ for which $p_n \neq q_n$ (as usual, we set $n(P,Q)=\infty$ if there is no such $n$). Define   
$$    
d(P,Q) = \frac{1}{n(P,Q)}.
$$
Show that $d$ is an ultra-metric on $X$ and the resulting metric space $(X,d)$ is complete. 

\end{document}

    
