Fredholm maps have countable proper local restrictions externally

Assume the Axiom of Choice (The Axiom of Choice). Let $f:M\to N$ be a $C^h$ map, $h\ge 1$, between Hausdorff second-countable real Banach manifolds (Countable base Banach manifold and smooth map), and suppose every $Df(x)$ has closed range and finite-dimensional kernel and cokernel. There is a finite or countable family of closed subsets $(C_j{)}_{j\in J}$ of $M$, indexed by $J\subseteq \mathbb{N}$, whose interiors cover $M$ such that, for every $j\in J$:

• $C_j$ is contained in a Fredholm normal-form neighbourhood $W_j$;
• $f{\mathrm{\mid }}_{C_j}:C_j\to N$ is proper, meaning inverse images of compact sets are compact; and
• $f(W_j)$ lies in a target chart.

Here a Fredholm normal-form neighbourhood means source and target coordinates in which

with $u$ in a Banach range space, $v$ in a finite-dimensional kernel space, and $g$ valued in a finite-dimensional obstruction space, as in Local finite-dimensional reduction for a Fredholm map. The chart may be shrunk before choosing the subordinate closed proper restriction. No closed ball in an infinite-dimensional Banach space is asserted compact. If $M$ is empty, take $J=\mathrm{\varnothing }$; no normal-form neighbourhood or target chart then needs to be chosen.

This countable localization and local-properness package is recorded from Smale and is not proved locally here.