$X$ compact Hausdorff space whose diagonal is a zero set: prove that the diagonal in $X \times X$ is a $G_{\delta}$ -set

Question

The problem says:

Suppose that $X$ is a compact Hausdorff space. If there exists a continuous function $f \colon X \times X \to \mathbb{R}$ such that $f (x, y) = 0 \iff x = y$, prove that the diagonal $\Delta$ in $X \times X$ is a $G_\delta$-set, and hence deduce that $X$ is second countable. (Thus $X$ is metrisable.)

A subset $A$ of a space $X$ is called $G_\delta$ if it is the intersection of at most countably many open sets

What I'm doing is evaluating the diagonal in the function, which gives me the diagonal equals zero. Now zero is $G_\delta $ on the real axis. And I do not know how to continue from here.

Answer 1

$$\Delta=f^{-1}(0)=f^{-1}\Big( \bigcap_{n \in \mathbb{Z}_+}(-\frac{1}{n},\frac{1}{n})\Big)=\bigcap_{n \in \mathbb{Z}_+}f^{-1}((-\frac{1}{n},\frac{1}{n})),$$ and each $f^{-1}((-\frac{1}{n},\frac{1}{n}))$ is open in $X\times X$ by continuity of $f.$