PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)

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The critical value set of a smooth map is sigma-compact

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For a smooth map $F : M \to N$ , the critical value set $CV (F)$ is a $σ$ -compact subset of $N$ .

Facts & Assumptions

Given: ${A C}_{ω}$ and a smooth map $F : M \to N$ .

[F1]

The critical value set is the image of the critical locus (The critical locus and critical value set).

[L1]

The submersion locus is open, so the critical locus is closed; under ${A C}_{ω}$ every manifold has a compact exhaustion; and continuous images of compact sets are compact (The immersion and submersion loci are open, Every manifold has a compact exhaustion, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset).

Proof

technique · direct

1.1L1givenchoose

By [L1], the critical locus $Crit (F)$ is closed in $M$ . Under the given ${A C}_{ω}$ , let $K_{1} \subseteq K_{2} \subseteq \dots$ be a compact exhaustion of $M$ from [L1]. This is the only use of countable choice.

2.1L1step 1.1

Then $Crit (F) \cap K_{j}$ is compact for every $j$ , so [L1] makes $F (Crit (F) \cap K_{j})$ compact in $N$ .

3.1F1step 2.1algebra∎

By [F1], $CV (F) = ⋃_{j \geq 1} F (Crit (F) \cap K_{j}),$ so $CV (F)$ is a countable union of compact sets.

Depends on

Used by

Regular values form a dense G_δ set Corollary

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)