ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Rank-zero and empty-base vector-bundle classification

Example

For every space $X$ , the rank-zero vector bundle is, up to its unique bundle isomorphism, $X \overset{i d}{\to} X$ . Thus ${Vect}_{0}^{F} (X)$ and $[X, {Gr}_{0} (F^{\infty})]$ are singletons. If $X = \emptyset$ and $n$ is arbitrary, the unique empty total-space bundle and the unique map from the empty space likewise give singleton classification sets.

Facts & Assumptions

Given: a space $X$ , a field $F \in {R, C}$ , and a nonnegative integer $n$ .

[F1]

A rank- $n$ vector bundle is locally a projection $U \times F^{n} \to U$ (Real and complex topological vector bundles).

[F2]

${Gr}_{0} (F^{N})$ and ${Gr}_{0} (F^{\infty})$ are one-point spaces carrying the zero tautological bundle (Stiefel spaces, Grassmannians, and tautological bundles).

Verification

technique · direct

1.1

Let $p : E \to X$ have rank zero. Every fiber is the one-element vector space $F^{0} = {0}$ by [F1], so $p$ is bijective. Each bundle chart is a homeomorphism $p^{- 1} (U) ≅ U \times {0} ≅ U$ over $U$ , and these local inverses glue to the inverse of $p$ . Thus $p$ is a bundle isomorphism to ${i d}_{X} : X \to X$ , and any bundle map over $X$ between two such bundles is forced fiberwise. There is exactly one rank-zero isomorphism class.

F1construct

2.1

By [F2], there is exactly one map $X \to {Gr}_{0} (F^{\infty})$ and exactly one homotopy class of such maps. Pulling back the zero tautological bundle gives the bundle in step 1.1, so the two singleton sets correspond.

F2step 1.1

3.1

Now let $X = \emptyset$ and allow any $n$ . A map $E \to \emptyset$ exists only when $E = \emptyset$ , and local triviality is vacuous, so this is the unique rank- $n$ bundle. There is also exactly one function from $\emptyset$ to ${Gr}_{n} (F^{\infty})$ and exactly one homotopy between any two such functions. Hence both classification sets are again singletons. No choice principle is used.

F1F2construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Hatcher, Vector Bundles & K-Theory, Section 1.1 conventions (standard reference, not scraped)