Let $C[0,1]$ be the $C^*$-algebra of continuous functions $[0,1]\to\mathbb C$.
I understand that the closed (two-sided, $*$-closed) ideals in $C[0,1]$ are in correspondence with the closed subspaces of $[0,1]$, with the correspondence is given by assigning to a subspace the set of functions that vanish on it.
What happens if one wants to look at the set of all (not-necessarily closed) ideals in $C[0,1]$, which we now view just as a commutative $\mathbb C$-algebra? From this post, I see that such ideals do exist.
Question 1: Does the set of not necessarily closed ideals in $C[0,1]$ have an easy description?
Question 2: More generally, does there exist such a description for $C_0(X)$ for some locally compact Hausdorff space $X$?
Comment: I would guess that if $I$ is an ideal (not necessarily closed) of $C_0(X)$, there would have to be a point in $X$ at which all functions in $I$ vanish simultaneously, but it's just a guess.