(a) Let $(X, d)$ be a metric space containing the point $x_{0}$, and let

$$U=\left\{x \in X: d\left(x, x_{0}\right)<1\right\}, \quad K=\left\{x \in X: d\left(x, x_{0}\right) \leqslant 1\right\}$$

Is $U$ necessarily the largest open subset of $K$ ? Is $K$ necessarily the smallest closed set that contains $U$ ? Justify your answers.

(b) Let $X$ be a normed space with norm $\|\cdot\|$, and let

$$U=\{x \in X:\|x\|<1\}, \quad K=\{x \in X:\|x\| \leqslant 1\}$$

Is $U$ necessarily the largest open subset of $K$ ? Is $K$ necessarily the smallest closed set that contains $U$ ? Justify your answers.