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 manifold bump for a compact set inside an open set

Statement

Let $M$ be a smooth manifold, let $K \subseteq M$ be compact, and let $W \subseteq M$ be open with $K \subseteq W$ . Then there exists a smooth function $ρ : M \to [0, 1]$ that equals $1$ on an open neighbourhood of $K$ and satisfies $supp (ρ) \subseteq W$ .

Facts & Assumptions

Given: A compact set $K \subseteq M$ and an open set $W \subseteq M$ with $K \subseteq W$ .

[L1]

Every point of $K$ admits a smooth bump supported in $W$ and equal to $1$ at that point (A chart bump at a point with prescribed support).

[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 a smooth manifold are smooth.

Proof

technique · direct

1.1

For each $p \in K$ , choose a smooth bump $ρ_{p}$ from [L1] with support in $W$ and $ρ_{p} (p) = 1$ ; compactness gives finitely many points $p_{1}, \dots, p_{m}$ such that the open sets $V_{p_{i}} : = ρ_{p_{i}}^{- 1} ((1 / 2, 1])$ cover $K$ .

L1givenchoose

2.1

Put $s : = ρ_{p_{1}} + \dots + ρ_{p_{m}}$ ; then $s$ is smooth by [A1], one has $2 s \geq 1$ on the open neighbourhood $V_{p_{1}} \cup \dots \cup V_{p_{m}}$ of $K$ , and $supp (s) \subseteq W$ .

A1step 1.1

3.1

Define $ρ : = σ \circ (2 s)$ ; then $ρ$ is smooth, equals $1$ on an open neighbourhood of $K$ , and vanishes off $W$ , so $supp (ρ) \subseteq W$ .

F1step 2.1∎

Depends on

Used by

Dependency tree · two levels

8 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)