Closure properties

Geometry

Here are three statements about the closure $\overline S$ of a set $S$ inside a metric space $X.$

I. Let $S,T$ be subsets of $X.$ Then $\overline{S \cap T} = {\overline S} \cap {\overline T}.$
II. Let $S,T$ be subsets of $X.$ Then $\overline{S \cup T} = {\overline S} \cup {\overline T}.$
III. If ${\overline S} = X,$ then $S=X.$

Which of these statements is/are true?