<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 \b[functional separation of <closed sets>], stronger than merely choosing disjoint open <neighbourhoods>.

Solution

Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / A / 2 / i / Solution

 Solution (source code)