 Solution

Urysohn's lemma states that if $X$ is normal and $A,B$ are disjoint closed subsets, there is a continuous function $u:X\to [0,1]$ with $u{|}_A=0$ and $u{|}_B=1$. The values on the rest of $X$ remain in that interval. Interchanging $A,B$ or composing with an affine function gives any two prescribed real endpoint values. This is functional separation of closed sets, stronger than merely choosing disjoint open neighbourhoods.