Suppose \(A\subseteq X\) is a connected subset. Then, the closure \(\bar{A}\) is also connected. Moreover, every set \(B\subseteq X\) satisfying \(A\subseteq B\subseteq \bar{A}\) is also connected.

Proof
Since \(A\) is dense in \(\bar{A}\), (0x681ad42d) implies that \(\bar{A}\) is connected. The same is true for \(B\) satisfying the mentioned conditions.