The Cantor set is closed and has empty interior

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 a closed subset of \(\RR\) with empty interior.