(a) Show that every compact subset of a Hausdorff topological space is closed.

(b) Let $X$ be a compact metric space. For $F$ a closed subset of $X$ and $p$ any point of $X$, show that there is a point $q$ in $F$ with

$$d(p, q)=\inf _{q^{\prime} \in F} d\left(p, q^{\prime}\right)$$

Suppose that for every $x$ and $y$ in $X$ there is a point $m$ in $X$ with $d(x, m)=(1 / 2) d(x, y)$ and $d(y, m)=(1 / 2) d(x, y)$. Show that $X$ is connected.