$(X,\mathscr T)$ is normal and each closed subsets of $X$ is a $G_{\delta}$ set.Then, $(X,\mathscr T)$ perfectly normal.

Question

Prove the following result without using Urysohn's lemma.

$(X,\mathscr T)$ is normal and every closed subset of $X$ is a $G_{\delta}$ set.Then, $(X,\mathscr T)$ perfectly normal.

My effort: I have proved using Urysohn's lemma. How do I prove without the use of Urysohn's lemma?

Let $A$ and $B$ be disjoint closed sets of $X$. My aim is to construct a continuous function from $X\to [0,1]:f^{-1}{(0)}=A$ and $f^{-1}{(1)}=B.$ Using Urysoh's lemma resultfollows trivially.

Answer 1

This question is very similar, maybe you're using the same text?

As I said there: you really cannot avoid Urysohn's lemma there, or you'd have to reprove it, with extra modifications. What other ways do you have to construct functions on a normal space?