Let \(A\subseteq B \nsubseteq X\). The set \(A\) is dense in \(X\) if and only if \(A\) is dense in \(B\) and \(B\) is dense in \(X\).

Proof

Assume \(A\) is dense in \(B\) and \(B\) is dense in \(X\). Then

\[ X=\bar{B}=\overline{B\cap A}\subseteq \bar{B}\cap \bar{A}=X\cap \bar{A}. \]

The other direction follows directly by definition.