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.

Real projective space from affine charts

Example

Real projective space ${R P}^{n}$ is the quotient of $R^{n + 1} ∖ {0}$ by the relation $x \sim λ x$ for $λ \neq 0$ , with the quotient topology (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection). For $0 \leq i \leq n$ let

$U_{i} : = {[x_{0} : \dots : x_{n}] \in {R P}^{n} : x_{i} \neq 0} .$

The affine coordinate map

$ϕ_{i} ([x_{0} : \dots : x_{n}]) = (\frac{x_{0}}{x_{i}}, \dots, \hat{\frac{x_{i}}{x_{i}}}, \dots, \frac{x_{n}}{x_{i}}) \in R^{n}$

defines a chart on $U_{i}$ , and these charts form a smooth atlas.

Facts & Assumptions

Given: The quotient model of ${R P}^{n}$ , the open sets $U_{i}$ , and the affine coordinate maps $ϕ_{i}$ .

[F1]

The quotient topology is the one for which a subset is open exactly when its full preimage is open (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[F2]

A smooth atlas is a covering family of pairwise smoothly compatible charts (Smooth atlases).

[F3]

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

Verification

technique · direct

1.1F1

The sets $U_{i}$ cover ${R P}^{n}$ because every nonzero vector in [F1] $R^{n + 1}$ has at least one nonzero coordinate. Each $U_{i}$ is open by [F1], since its preimage is ${x_{i} \neq 0} \subseteq R^{n + 1} ∖ {0}$ . The inverse chart sends $(u_{1}, \dots, u_{n}) \in R^{n}$ to the projective class with $i$ -th coordinate $1$ and the remaining coordinates given by the $u_{j}$ , so each $ϕ_{i}$ is a homeomorphism $U_{i} \to R^{n}$ .

2.1F2step 1.1

On $U_{i} \cap U_{j}$ the transition map $ϕ_{j} \circ ϕ_{i}^{- 1}$ is obtained by [F2, step 1.1] dividing all affine coordinates by the coordinate corresponding to $x_{j} / x_{i}$ , which is nonzero on the overlap. Thus every transition function is rational with nonvanishing denominator on its domain, hence smooth. Therefore the family $(U_{i}, ϕ_{i})$ is a smooth atlas by [F2].

3.1F3step 2.1∎

This atlas equips ${R P}^{n}$ with a smooth structure, so [F3] makes [F3, step 2.1] ${R P}^{n}$ a smooth $n$ -manifold.

Depends on

Used by

Dependency tree · two levels

12 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 1.10 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)