The Cantor set is closed but contains no interval, so its interior is empty.Suggest a slogan
Let \(C_0\coloneqq[0,1]\), and let \(C_{k+1}\) be obtained from \(C_k\) by removing the open middle third of each of the \(2^k\) disjoint closed intervals of length \(3^{-k}\) whose union is \(C_k\). The Cantor set \(C\coloneqq\bigcap_{k\geq 0}C_k\) is nowhere dense in \(\RR\).
Each \(C_k\) is closed, so \(C\) is closed and equals its closure. If \(C\) contained an open interval \(I\) of length \(\ell>0\), then \(I\subseteq C_k\) for every \(k\); since \(I\) is connected, it would lie in one of the intervals of length \(3^{-k}\) making up \(C_k\), so \(\ell\leq 3^{-k}\) for every \(k\), which is impossible. Hence the closure of \(C\) has empty interior.