Critical images of proper local Fredholm restrictions are nowhere dense externally

Assume the Axiom of Choice (The Axiom of Choice). Let $f:M\to N$ be a $C^h$ map of fixed Fredholm index $m$ between Hausdorff second-countable real Banach manifolds (Countable base Banach manifold and smooth map), where $h$ is a positive integer or $\mathrm{\infty }$ and $h>\max \{m,0\}$. Suppose $W\subseteq M$ is open, $f(W)$ lies in a target chart, and on $W$ the map has Fredholm normal form

where the kernel variable lies in a finite-dimensional space $K$, the obstruction component lies in a finite-dimensional space $Q$, and $\dim K-\dim Q=m$ (Local finite-dimensional reduction for a Fredholm map).

If $C\subseteq W$ is closed in $M$ and $f{\mathrm{\mid }}_C:C\to N$ is proper, then

is closed and nowhere dense in $N$, where $\mathrm{Crit}(f)=\{x\in M:Df(x)\text{ is not surjective}\}$ and nowhere dense has the meaning in Nowhere dense, meagre, residual, and comeagre subsets of a topological space.

This is the localized category consequence used in Smale's proof, not a separately numbered theorem there and not a local proof in this library.