ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-29

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.

Euclidean spaces and Euclidean open subsets as smooth manifolds

Example

For every $n \geq 0$ , the Euclidean space $R^{n}$ is a smooth $n$ -manifold, with global chart the identity map. More generally, every open subset $U \subseteq R^{n}$ is a smooth $n$ -manifold with its standard restricted smooth structure.

Facts & Assumptions

Given: A natural number $n$ and an open subset $U \subseteq R^{n}$ .

[F1]

Open subsets of Euclidean space carry the standard smooth structure (Open subsets of Euclidean space have the standard smooth structure).

[F2]

A smooth manifold is a topological manifold equipped with a smooth structure (Smooth manifolds and their smooth charts).

Verification

technique · direct

1.1F1F2

Taking $U = R^{n}$ , the identity chart exhibits $R^{n}$ as a topological $n$ -manifold and [F1] supplies its smooth structure. Hence $R^{n}$ is a smooth $n$ -manifold by [F2].

2.1F1F2∎

For a general open subset $U \subseteq R^{n}$ , [F1] states exactly that $U$ inherits the standard smooth structure, so again [F2] makes $U$ a smooth $n$ -manifold.

Depends on

Used by

Geometric cardinality is not homotopy invariant Counterexample
Noncompact intersections can escape during a homotopy Counterexample
Two noncompatible atlases on the real line Counterexample
A circle is the boundary of a disk Example
A countable disjoint union of lines is a smooth manifold Example
A nonsingular affine conic is a punctured-plane Riemann surface Example
Atlases on the sphere, plane, disc and annulus Example
Distance hessian in euclidean space Example
Nonsingular affine and projective curves as Riemann surfaces Example
The complex torus as a Riemann surface Example
The pair of pants is a cobordism realizing addition of circles Example
The real affine group is amenable and nonunimodular Example
Two projective lines have one mod 2 intersection Example
Two unoriented points bound an interval Example
Every integer is realized by a map to the sphere Lemma
Oppositely framed points cancel in pairs Lemma
Smooth relative isotopy extension for disk arcs with puncture endpoints Lemma
Smooth relative isotopy extension for finite disk arc systems Lemma
Zero-dimensional bordism groups Proposition
Cut locus of a point has riemannian volume zero Theorem
Framed zero-dimensional bordism in a nonorientable manifold is mod two Theorem
Framed zero-dimensional bordism in an oriented manifold is the integers Theorem

Dependency tree · two levels

11 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

Rob van der Vorst, Introduction to differentiable manifolds, §1, Example 1.5 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)