 Zero set of a continuous function

For a continuous function $f:X\to \mathbb{R}$, its zero set is $f^{-1}(\{0\})$. It is closed and is a G-delta set, since $f^{-1}(0)={\bigcap }_{n\ge 1}{f}^{-1}((-1/n,1/n))$. Replacing $f$ by $|f|/(1+|f|)$ gives the same zero set with values in $[0,1]$. In a normal topological space, every closed G-delta set has this form.