show that a perfectly normal space is completely normal

Question

The problem I want to ask is in the post title. It's in Munkres books. I type it one more time below for you

Show that a perfectly normal space is completely normal.

In perfectly normal space, every closed set is a $G_{\delta}$ set. Let $A$, $B$ be separated sets in $X$. Because $\bar{A}, \bar{B}$ are closed $G_{\delta}$ in normal space $X$, so we can find 2 continuous functions $f, g$ vanish precisely on $\bar{A}$ and $\bar{B}$ respectively. This is where I got stuck. Let $h = f - g$, then $h$ is continous. I need to find 2 disjoint open sets $C,D$ in $[-1,1]$ such that $h^{-1}(C)$ contains $A$ and $h^{-1}(D)$ contains $B$. But I can't find them.

Thanks everybody. I really I appreciate if some one can help me solve this.

Answer 1

(After taking note of my comment to the question above)

Hint: Since $A$ and $B$ are separated, then $h(x) < 0$ for each $x \in A$ and $h(x) > 0$ for each $x \in B$.

Answer 2

1. In a perfecly normal space $X$ all closed sets are $G_\delta$ and so all open sets are $F_\sigma$.

2. A space $X$ is completely normal iff all open subspaces are normal.

3. An $F_\sigma$ subspace of a normal space is normal.

For a proof of 3. see my note here or any decent textbook.

Together these imply what you want.