What can we say about a not-regular level sets of a smooth function?

Question

Suppose $F : M \to \mathbb{R}$ is a smooth function, $M$ a smooth manifold and $a \in \mathbb{R}$ is a critical value. Of course $N = F^{-1}(a)$ need not be an embedded submanifold of $M$.

1. How could I say that $N$ has some form of manifold structure, either immersed or even just a topological manifold?
2. What topological properties does $N$ have beyond merely being a closed set?
3. Are there any other properties of such level sets that you personally like whatsoever?