Smooth map on manifolds from a given closed subset to a given point

Question

I recently learned that if $M$ and $N$ are both manifolds, then for a smooth map $F:M\to N$, the preimage $F^{-1}(q)$ of a point $q\in N$ is always a closed subset of $M$. How about the other direction? If we are given a closed subset $S\subset M$ and a point $q\in N$, can we define a smooth map $F:M\to N$ such that $S=F^{-1}(q)$ explicitly? Thank you!

Answer 1

The answer is yes, if $\dim N > 0$ (as @SeverinSchraven points out in the comments) via the following result from Lee's Introduction to Smooth Manifolds:

Theorem 2.29 (Level Sets of Smooth Functions). Let $M$ be a smooth manifold. If $K$ is any closed subset of $M$, there is a smooth nonnegative function $f\colon M \to \mathbb{R}$ such that $f^{-1}(0) = K$.

Taking such an $f$ and composing it with a smooth map $i\colon \mathbb{R} \to N$ such that $i^{-1}(q) = \{0\}$ then yields the result.