Closure properties

Geometry Level 4

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 \(S=X.\)