= Solution
\b[Neither assertion holds for a general <metric space>.] Take $X=\{0,1\}$ with $d(0,1)=1$ and center $0$. This is a <discrete topology>: both singleton sets are open and closed. The <open ball> is $U=\{0\}$, whereas the <closed ball> is $K=X$. Thus the <interior> of $K$ is $X$, strictly larger than $U$, and the <closure> of $U$ is $\{0\}$, strictly smaller than $K$. In general $U\subseteq\operatorname{int}K$ and $\overline U\subseteq K$, but the inclusions need not be equalities.
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