In 1948 Whitney proved his ideal (spectral) theorem [1] describing the closed ideals
Let $M$ be an $n$-dimensional manifold. for each point $p \in M$ and each natural $k$ we define $N(k)$ to be the number of (up to $n$) tuples $m$ such that $|m| \leq k$. Define the map $J_p^k: C^\infty(M) \rightarrow \mathbb{R}^{N(k)}$ by assigning to $f$ the $m$-jets of $f$ at $p$ up to $|m|=k$.
If $I$ is an ideal of $C^\infty(M)$ then its closure is the ideal of functions $f$ such that for each $p$ in $M$ and $k \geq 0$ then $J^k_p f \in J^k_p(I)$.
So in some sense the closed ideals are like $I_\infty$ in Neil's answer.
[1] H.Whitney. On ideals of differentiable functions. American Journal of Mathematics. Vol. 70, No. 3, pp. 635-658 (1948)
