Perfectly normal space

A T1 space is perfectly normal when it is normal and every closed subset is a G-delta set. Equivalently every closed subset is the zero set of a continuous function into $[0,1]$, using the closed G-delta set as a zero set theorem. If disjoint closed sets are the exact zero sets of $f,g$, then $f/(f+g)$ has those sets as its exact zero and one levels. Every subspace is normal, although the converse fails.

For disjoint closed sets $A,B$ in a subspace $Y$, their ambient closures $C,D$ satisfy $(C\cap D)\cap Y=\varnothing$. In a perfectly normal space choose continuous functions $f,g$ with exact zero sets $C,D$. The function $f/(f+g)$ is continuous on $X\setminus (C\cap D)$, which contains $Y$, and separates $A,B$. Disjoint inverse images of endpoint neighborhoods give normality of $Y$.