 Closed-neighborhood shrinking in a normal space

If $F$ is closed, $U$ is open, and $F\subset U$ in a normal topological space, separate $F$ from the closed complement $X\setminus U$ by disjoint open sets $W,V$. Then $\overline{W}\subset X\setminus V\subset U$. In particular, first separate two disjoint closed sets by disjoint open sets and then shrink each neighborhood this way: their closures are disjoint. No additional metric or compactness assumption is needed.