Normality is exactly the ability to separate disjoint closed sets by a continuous $[0,1]$-valued function.Suggest a slogan
Let \(X\) be a topological space. Then \(X\) is normal if and only if for every pair of disjoint closed subsets \(A, B \subseteq X\) there is a continuous function \(f\colon X\to [0,1]\) with \(f(a) = 0\) for all \(a\in A\) and \(f(b) = 1\) for all \(b\in B\) [@Mun00].
The forward direction is Urysohn’s lemma; the converse follows by taking the disjoint open sets \(f^{-1}([0,1/2))\) and \(f^{-1}((1/2,1])\).
State Urysohn's lemma.
Which separation property does Urysohn's lemma characterise, and by what function?