LemmaStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-24 (gpt-6-sol)

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Countable proper local restrictions of a Fredholm map

Statement

Assume the Axiom of Choice. Let $P : M \to N$ be a $C^{h}$ Fredholm map, $h \geq 1$ , between Hausdorff second-countable real Banach manifolds. There is a countable family of subsets $C_{j} \subset M$ whose interiors cover $M$ . For each $j$ there are a source open set $W_{j}$ and a target chart $V_{j}$ such that $C_{j}$ is closed relative to $W_{j}$ , $P (W_{j}) \subset V_{j}$ , $P ∣_{C_{j}} : C_{j} \to V_{j}$ is proper, and in coordinates on $W_{j}, V_{j}$ the map has Fredholm normal form $(u, v) ⟼ (u, g_{j} (u, v)),$ where $v$ and the target obstruction coordinate are finite-dimensional. Proper means that the inverse image of each compact subset of $V_{j}$ is compact. The conclusion includes $M = \emptyset$ , with an empty family.

Facts & Assumptions

Given: AC and the map in the statement.

[F1]

Every point has a normal-form neighbourhood with product source coordinates $U \times A$ , where $U$ is open in a Banach range space and $A$ is open in a finite-dimensional kernel space (Local finite-dimensional reduction for a Fredholm map).

[F2]

The source is second countable (Countable base Banach manifold and smooth map).

Proof

technique · direct

1.1

If $M$ is empty, the empty family works. Otherwise fix $x \in M$ and use [F1] to obtain $W$ , a source coordinate diffeomorphism $T : W \to U \times A$ , and a target chart in which $P (T^{- 1} (u, v)) = (u, g (u, v))$ . Write $T (x) = (u_{x}, v_{x})$ . Choose radii $a, b > 0$ small enough that the closed balls ${\overline{B}}_{a} (u_{x}) \subset U$ and ${\overline{B}}_{b} (v_{x}) \subset A$ . Set $C_{x} : = T^{- 1} ({\overline{B}}_{a} (u_{x}) \times {\overline{B}}_{b} (v_{x})) .$ Then $C_{x}$ is closed relative to $W$ , and its interior in $M$ contains $x$ .

F1givenchoose

2.1

The restriction $P ∣_{C_{x}} : C_{x} \to V$ is proper. To see this, let $K \subset V$ be compact and take a sequence $(z_{i})$ in $C_{x} \cap P^{- 1} (K)$ . After a subsequence $P (z_{i})$ converges in $K$ ; in target coordinates its first components $u_{i}$ therefore converge to some $u \in {\overline{B}}_{a} (u_{x})$ . The $v_{i}$ lie in the compact finite-dimensional ball ${\overline{B}}_{b} (v_{x})$ , so a further subsequence converges to $v \in {\overline{B}}_{b} (v_{x})$ . Continuity of $T^{- 1}$ and $P$ gives $z_{i} \to T^{- 1} (u, v) \in C_{x} \cap P^{- 1} (K)$ . This subset is metrizable through $T$ , so sequential compactness implies compactness. The argument uses no compactness of a ball in the Banach range coordinate.

step 1.1algebra

3.1

The interiors of the $C_{x}$ form an open cover of $M$ . By [F2], under AC every open cover has a countable subcover: for each member of a countable base that is contained in some cover member, choose one such member. Keep the corresponding countably many $C_{x}$ , and index them by a subset of $N$ . Their interiors still cover $M$ , and each retains the properties proved in steps 1.1–2.1.

F2step 1.1step 2.1choose∎

Depends on

Used by

Sard--Smale residual regular values for Fredholm maps Theorem

Dependency tree · two levels

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Sources

Stephen Smale, An Infinite Dimensional Version of Sard's Theorem, proof of Theorems 1.3 and 1.6, pp. 862-863 (standard reference, not scraped)