False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 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-value set need not be closed

Statement

False claim: the critical value set of a smooth map is always closed.

Facts & Assumptions

Given: A smooth bump $β : R \to R$ supported in $[- 1, 1]$ with $β (0) = 1$ , $β^{'} (0) = 0$ , and $0 < β (t) < 1$ for $0 < ∣ t ∣ < 1$ , together with the smooth map

f (x) : = \sum_{k = 1}^{\infty} (1 - \frac{1}{k}) β (2^{k} (x - k)) .

[L1]

A sigma-compact set need not be closed, and regular values can be dense despite the presence of critical values accumulating at them (The critical value set of a smooth map is sigma-compact, Regular values form a dense $G_{δ}$ set).

Refutation

technique · direct

1.1

The supports of the summands are pairwise disjoint, so the series defines a smooth function. At each center $x = k$ , the derivative is zero and $f (k) = 1 - \frac{1}{k} .$ Thus every value $1 - \frac{1}{k}$ is a critical value.

givenalgebra

2.1

The sequence $1 - \frac{1}{k}$ converges to $1$ . But $f (x) \neq 1$ for every $x$ , because every summand has height strictly below $1$ and outside the supports the function is $0$ . Hence $1$ is a regular value with empty fibre, not a critical value.

step 1.1algebra

3.1

Therefore the critical value set contains ${1 - \frac{1}{k} : k \geq 1}$ but not its limit $1$ , so it is not closed. This is consistent with [L1].

L1step 2.1∎

Depends on

Used by

A smooth map with a nonclosed critical-value set Example

Dependency tree · two levels

7 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)