Is there an alternative proof to Urysohn's lemma, that makes use of $d(x, A)$?
Urysohn's lemma is: given a normal topological space $X$, for any disjoint closed sets $A$, $B$, there exists a separation by continuous function $f$. I.e. there exists $f : X \to [a,b]$ with $a,b \in \Bbb{R}$ such that $f$ is continuous and $f(A) = a$ while $f(B) = b$, for all $a,b \in \Bbb{R}$.
My question is, since $d(x,A), d(x,B)$ are continuous functions from $X$ into $\Bbb{R}$, can we somehow construct the required Urysohn function from this?
Thanks.