Confusion about this assertion

Question

I have seen this result true in general ?

Every zero set is $G_\delta$-sets.

As I know in Normal spaces, every closed $G_\delta$ set is zero set.

Thx in advance

Answer 1

Let $F$ be a zero set. By definition there is a continuous function $f:X\to\mathbb R$ so that $f^{-1}(0)=F$.

Note that $\{0\}=\bigcap_{n=1}^\infty (\frac{-1}{n},\frac{1}{n}),$ it follows that $$F=f^{-1}[\bigcap_{n=1}^\infty (\frac{-1}{n},\frac{1}{n})]=\bigcap_{n=1}^\infty f^{-1}[(\frac{-1}{n},\frac{1}{n})],$$ hence $F$ is a $G_\delta$-set.

Answer 2

Assume that $A=f^{-1}(0)$, where $f:X\to I$. Then $$A=\bigcap\limits_{n=1}^\infty f^{-1}\left(\left[0,\frac1n\right)\right)$$