TITLE: Is a closed $G_\delta$ set in a Hausdorff space always a zero set?
QUESTION [9 upvotes]: I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?

REPLY [3 votes]: Not even in completely regular Hausdorff spaces.  In general we have
$$
\text{compact $G_\delta$}\qquad\Longrightarrow\qquad
\text{zero-set}\qquad\Longrightarrow\qquad
\text{closed $G_\delta$}
$$
but none reversible.