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\lhead{\bf Advanced Calculus (MAA 4211)}
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\rhead{Due Wednesday 11/28}
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\begin{enumerate}

\Problem Let $X$ be an infinite set and $d$ be the discrete metric. Determine, with proof, the connected sets in the metric space $(X,d)$.

\Problem Let $(X,d)$ be an arbitrary metric space. Is the intersection of two connected sets in $(X,d)$ necessarily connected? \emph{Prove or give a counterexample.}

\Problem Prove that if $S$ is a connected subset of the metric space $(X,d)$ then its closure $\overline{S}$ is also connected.

\Problem Let $(X,d_X)$ be any metric space and $(Y,d_Y)$ be any metric space \emph{with the discrete metric}. Determine, with proof,
\begin{enumerate}
\item[(a)] all continuous functions $f : Y\to X$ and
\item[(b)] all continuous functions $f : \mathbb{R}\to Y$, where the metric on $\mathbb{R}$ is the usual Euclidean one.
\end{enumerate}

\end{enumerate}

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