DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The critical locus and critical value set

Definition

Let $F : M \to N$ be smooth. The critical locus of $F$ is

$Crit (F) : = {p \in M : p is a critical point of F},$

and the critical value set of $F$ is its image

$CV (F) : = F (Crit (F)) .$

Its complement $N ∖ CV (F)$ is the set of regular values.

Depends on

Regular and critical points and values

Used by

Regular values have null complement and are dense Corollary
A constant map has a large critical locus and one critical value Example
A smooth map with a nonclosed critical-value set Example
Critical points and values of a height function on a sphere Example
Critical points need not be isolated False statement
The critical value set of a smooth map is sigma-compact Proposition
Morse-Sard for Euclidean maps Theorem
Morse-Sard for smooth manifolds Theorem

Dependency tree · two levels

2 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)