Example of closed set that is not a zero-set

Question

Is known that every zero-set (preimage of 0 by a continuous function) is closed, but the reverse is true (i read) just for perfectly normal spaces. I'm looking for a (i think so it should be not perfectly normal) space $X$ in wich there is a closed set that is not equal to the preimage of 0 of any continuous function from $X$ to $[0,1]$ and a specific example of such a set.

Thanks in advance

Answer 1

One simple example would be the line with two origins.

The singleton of one of the of the origins is closed but not a zero set.

Answer 2

Consider the space $X=\{0,1\}^I$ with $I$ uncountable, in the product topology. This is a compact Hausdorff (hence normal) space such that every singleton $\{x\}$ with $x \in X$ is closed and not a $G_\delta$, so it cannot be a zero-set (as all zero sets $f^{-1}[\{0\}]$ are also of the form $\cap_n f^{-1}[(-\frac1n,\frac1n)]$, which is a $G_\delta$).