Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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  • 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.
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The pullback bundle is the set-theoretic inverse image of the total space

Statement

The pullback bundle is the set-theoretic inverse image of the original total space.

Facts & Assumptions

Given: The displayed claim.

[L1]

The pullback bundle consists of pairs (q,e) with f(q)=π(e) (Pullback vector bundles as fibre products).

Refutation

technique · direct
1.1

Let f:R→{0}⊆R be the constant map and let E=R×R→R be the trivial line bundle. Then f∗E={(q,(0,v)):q∈R, v∈R}, which is naturally R×R.

L1givenconstruct
2.1

Different base points q1≠q2 with the same fibre element v give distinct pullback points (q1,(0,v))≠(q2,(0,v)). Thus the pullback keeps new base information and is not a subset of the old total space. It is a fibre product, not a set-theoretic inverse image.

L1step 1.1algebra∎

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