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\noindent \underline{Homework 5 -- MATH 4590 Spring 2018} \\

\begin{enumerate}
\item Let $(M,d)$ be a metric space and let $U \subset M$. Prove that $U$ is open if and only if none of its points are a limit of its complement.

\item Prove that the union of finitely many closed sets is a closed set.

\item Consider $(M,d)=(\mathbb{R}^2,d)$, where $d$ is the Euclidean metric. We define the distance between two nonempty sets $A, B \subset \mathbb{R}^2$ by the following:
$$\mathrm{dist}(A,B)=\inf \{ d(a,b) \colon a \in A, b \in B\}.$$
Give an example of two nonempty \textit{disjoint} sets $A, B \subset \mathbb{R}^2$ such that $\mathrm{dist}(A,B)=0$.
\end{enumerate}

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