If $(X,d)$ complete and $X= \cup_{n=1}^{\infty} F_n$ where $F_n$ is closed in $X$ for all $n$, then there exist at least one $F_k$ which the interior is non empty.
The proof start by setting $U_n = X \backslash F_n$ open for all $n$. By the DeMorgan Law, $$\cap_{n=1}^{\infty}U_n=(\cup_{n=1}^{\infty} F_n)^c=X \backslash \cup_{n=1}^{\infty} F_n = \emptyset$$
Then it follow that at least one of the open $U_k$ is not dense in $X$ . I dont understand this part of the proof.
Can anyone help me please? thx.
A dense subset meets every non-empty open set, thus for the intersection to be empty at least one of the $U_n$ has to be not dense.