TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

Numerable vector bundles admit bundle metrics

Statement

Assume the Axiom of Choice. Every numerable real vector bundle has a continuous positive-definite fiber inner product, and every numerable complex vector bundle has a continuous Hermitian metric. Consequently this holds for bundles over paracompact Hausdorff bases.

AC supplies the dependent-choice consequence required by the published partition theorem. Once numerating charts and their subordinate partition are supplied, the metric construction is choice-free.

Facts & Assumptions

Given: AC and a finite-rank real or complex vector bundle $E \to X$ .

[F1]

A numeration consists of linear charts $E ∣_{U_{i}} ≅ U_{i} \times F^{n}$ and a locally finite partition $(ρ_{i})$ with $supp ρ_{i} \subseteq U_{i}$ (Real and complex topological vector bundles, Locally finite partitions of unity and subordination to an open cover).

[F2]

Under AC and DC, every open cover of a paracompact Hausdorff space has a subordinate locally finite partition of unity (Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity).

[A1]

AC is the stated choice principle, and it implies DC (The Axiom of Choice, AC supplies the dependent-choice instances used in vector-bundle constructions).

Proof

technique · direct

1.1

First suppose the numeration in [F1] is supplied. Transport the standard Euclidean or Hermitian form to $E ∣_{U_{i}}$ and call it $h_{i}$ . Define on each fiber $h_{x} (v, w) = \sum_{i} ρ_{i} (x) h_{i, x} (v, w)$ , taking the $i$ th term to be zero off $U_{i}$ . Since $supp ρ_{i} \subseteq U_{i}$ , this zero extension is continuous near every point outside $U_{i}$ , and local finiteness makes the sum continuous.

2.1

Each summand is positive semidefinite. At every $x$ , some $ρ_{i} (x) > 0$ because the coefficients sum to one; for $v \neq 0$ , the corresponding $h_{i, x} (v, v) > 0$ . Hence $h_{x} (v, v) > 0$ . The formula is symmetric bilinear over $R$ or conjugate-symmetric sesquilinear over $C$ , so it is the required metric.

F1step 1.1algebra

3.1

If $X$ is paracompact Hausdorff, apply [A1] to obtain DC and then [F2] to the linear chart cover of $E$ . This supplies a numeration, so steps 1.1–2.1 give a metric. AC is used only through this invocation of the published partition theorem; with supplied data those two steps make no choices.

F2A1step 1.1step 2.1∎

Depends on

Used by

Short exact sequences of numerable vector bundles split Corollary
Vector-bundle classification can fail without numerability Counterexample
General Thom isomorphism from the relative Serre spectral sequence Lemma
Thom isomorphism for oriented vector bundles Theorem

Dependency tree · two levels

18 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, Proposition 1.2 (standard reference, not scraped)
MIT 18.906 notes, Lecture 16 (standard reference, not scraped)