Prime ideals of set of continuous real valued functions on a compact hausdorff space

Question

Let $X$ be a compact Hausdorff space and $\mathcal P$ be a prime ideal of the ring of all real valued continuous functions on $X$. Then $\mathcal P$ must contained in one maximal ideal i.e. there is a point $y\in X$ such that each element of $\mathcal P$ vanishes at the point $y$. Does this imply that, $\{ y\}$ is $G_{\delta}$ in $X$?

Answer 1

No. Why would it imply $\{y\}$ is $G_\delta$ in $X$? For a simple example where it is not, let $X$ be any product of uncountably many compact Hausdorff spaces with more than one point. Then since any nonempty open set in the product topology is restricted on only finitely many coordinates, any nonempty $G_\delta$ set is restricted on only countably many coordinates, so no singleton is $G_\delta$.