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{\bf Supplementary problems (assigned 9/12/03)}
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\noindent We define the distance between any two non-empty subsets $A$ and $B$ 
of a metric space $X$ by 
$$
\dist(A,B) \eqdef \inf\{d(x,y):x\in A, y\in B\}.
$$
(Note that the infimum exists since $0$ is a lower bound for the set on
the right side.)  
\begin{enumerate}
\item Prove that $\dist(A,B) > 0$ implies $A\cap B=\emptyset$ (not much to
do here).
\item Give an example that shows we can have $\dist(A,B) = 0$ even
if $A\cap B = \emptyset$.
\item Prove that if $A$ is closed, $B$ is compact and $A\cap B = \emptyset$,
then $\dist(A,B) > 0$.
\end{enumerate}

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