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 \(S=X.\)

Which of these statements is true?

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