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 Euclidean bump for a compact set inside an open set

Statement

If $K \subseteq U \subseteq R^{n}$ with $K$ compact and $U$ open, then there exists a smooth function $ρ : R^{n} \to [0, 1]$ such that $ρ = 1$ on $K$ and $supp (ρ) \subseteq U$ .

Facts & Assumptions

Given: A compact set $K \subseteq R^{n}$ and an open set $U \supseteq K$ .

[L1]

For every $p \in K$ there are radii $0 < r_{p} < R_{p}$ with ${\overline{B}}_{r_{p}} (p) \subseteq B_{R_{p}} (p) \subseteq U$ .

[L2]

Each such concentric pair admits a smooth bump equal to $1$ on the inner closed ball and supported in the outer ball (A smooth bump between concentric Euclidean balls).

[F1]

The standard smooth step function $σ$ is $0$ on $(- \infty, 0]$ and $1$ on $[1, \infty)$ (The standard smooth step function).

[A1]

Finite sums of smooth real-valued functions on $R^{n}$ are smooth.

Proof

technique · direct

1.1

For each $p \in K$ , choose radii as in [L1] and a bump $ρ_{p}$ as in [L2]; compactness gives finitely many points $p_{1}, \dots, p_{m}$ such that $K \subseteq ⋃_{i = 1}^{m} B_{r_{p_{i}}} (p_{i})$ .

L1L2givenchoose

2.1

Put $s : = ρ_{p_{1}} + \dots + ρ_{p_{m}}$ ; then $s$ is smooth by [A1], one has $s \geq 1$ on $K$ , and $supp (s) \subseteq U$ .

A1step 1.1

3.1

Define $ρ : = σ \circ s$ ; then $ρ$ is smooth, equals $1$ on $K$ , and vanishes off $U$ , so $supp (ρ) \subseteq U$ .

F1step 2.1∎

Depends on

Used by

A chart bump at a point with prescribed support Lemma

Dependency tree · two levels

5 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)