LemmaStatement: 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.

A chart bump at a point with prescribed support

Statement

Let $M$ be a smooth manifold, let $p \in M$ , and let $W \subseteq M$ be open with $p \in W$ . Then there exists a smooth function $ρ : M \to [0, 1]$ such that $ρ (p) = 1$ and $supp (ρ) \subseteq W$ .

Facts & Assumptions

Given: A smooth manifold $M$ , a point $p \in M$ , and an open neighbourhood $W$ of $p$ .

[F1]

Smooth charts are diffeomorphisms onto open subsets of Euclidean space (Chart maps are diffeomorphisms onto Euclidean open sets).

[L1]

Compact sets inside Euclidean open sets admit smooth bumps with prescribed support (A Euclidean bump for a compact set inside an open set).

[L2]

Smooth maps that agree on overlaps paste to a smooth global map, and composites of smooth maps are smooth (Smooth maps paste over an open cover, Identity maps and composites of smooth maps are smooth).

[A1]

Closed bounded subsets of Euclidean space are compact, and compact subsets of the Hausdorff manifold $M$ are closed.

Proof

technique · direct

1.1

Choose a smooth chart $(U, φ)$ with $p \in U$ and put $a : = φ (p)$ . Choose $0 < r < R$ such that ${\overline{B}}_{r} (a) \subseteq B_{R} (a) \subseteq {\overline{B}}_{R} (a) \subseteq φ (W \cap U) .$ Applying [L1] to ${\overline{B}}_{r} (a) \subseteq B_{R} (a)$ gives a smooth $\tilde{ρ} : R^{n} \to [0, 1]$ equal to $1$ on ${\overline{B}}_{r} (a)$ and supported in $B_{R} (a)$ . Its support is compact by [A1].

F1L1A1givenchoose

2.1

Let $K : = φ^{- 1} (supp (\tilde{ρ}))$ . By step 1.1, $K$ is a compact, hence closed, subset of $W \cap U$ . On the open cover $U \cup (M ∖ K)$ , define $ρ = \tilde{ρ} \circ φ$ on $U$ and $ρ = 0$ on $M ∖ K$ . The formulas agree on $U ∖ K$ , so [L2] gives a smooth global function.

F1L2A1step 1.1

3.1

One has $ρ (p) = \tilde{ρ} (a) = 1$ , and $ρ$ vanishes outside $W$ , so $supp (ρ) \subseteq W$ .

step 1.1step 2.1∎

Depends on

Used by

A manifold bump for a compact set inside an open set Lemma

Dependency tree · two levels

20 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)