CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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.

Smooth functions separate points from closed sets

Statement

Let $A$ be a closed subset of a smooth manifold $M$ and let $p \in M ∖ A$ . Then there exists a smooth function $f : M \to [0, 1]$ such that $f (p) = 1$ and $f ∣_{A} = 0$ .

Facts & Assumptions

Given: A closed set $A \subseteq M$ and a point $p \in M ∖ A$ .

[L1]

A closed set inside an open set admits a smooth cutoff equal to $1$ near the closed set and supported in the open set (A smooth Urysohn lemma for a closed set in an open set).

Proof

technique · direct

1.1

The singleton ${p}$ is closed and lies in the open set $M ∖ A$ .

given

2.1

Apply [L1] to the closed set ${p} \subseteq M ∖ A$ ; the resulting function is $1$ at $p$ and vanishes on $A$ because its support lies in $M ∖ A$ .

L1step 1.1

3.1

This is the required separating smooth function.

step 2.1∎

Depends on

A smooth Urysohn lemma for a closed set in an open set

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)