LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Equality on a smaller neighbourhood defines the germ equivalence relation

Statement

Let $F$ be a presheaf on a topological space $X$ and let $x \in X$ . For pairs $(U, s)$ and $(V, t)$ with $x \in U \cap V$ , define $(U, s) \sim_{x} (V, t)$ when there exists an open neighbourhood $W$ of $x$ with $W \subseteq U \cap V$ and $s ∣_{W} = t ∣_{W} .$ Then $\sim_{x}$ is an equivalence relation.

Facts & Assumptions

Given: A presheaf $F$ on $X$ and a point $x \in X$ .

[F1]

The stalk is built from pairs $(U, s)$ modulo equality on a smaller neighbourhood of $x$ (The stalk of a presheaf at a point).

[F2]

An equivalence relation is one that is reflexive, symmetric, and transitive (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

Proof

technique · direct

1.1

Reflexivity holds because if $(U, s)$ is any pair, then $s ∣_{U} = s ∣_{U}$ , so $(U, s) \sim_{x} (U, s)$ . Symmetry holds because $s ∣_{W} = t ∣_{W}$ implies $t ∣_{W} = s ∣_{W}$ on the same neighbourhood $W$ .

F1F2

1.2

Suppose $(U, s) \sim_{x} (V, t)$ and $(V, t) \sim_{x} (Z, r)$ . Choose neighbourhoods $W_{1} \subseteq U \cap V$ and $W_{2} \subseteq V \cap Z$ with $s ∣_{W_{1}} = t ∣_{W_{1}}$ and $t ∣_{W_{2}} = r ∣_{W_{2}}$ . Then $W : = W_{1} \cap W_{2}$ is an open neighbourhood of $x$ contained in $U \cap Z$ , and on $W$ one has $s ∣_{W} = t ∣_{W} = r ∣_{W}$ . Hence $(U, s) \sim_{x} (Z, r)$ .

F1given

2.1

Steps 1.1 and 1.2 show that $\sim_{x}$ is reflexive, symmetric, and transitive. By [F2], it is an equivalence relation.

F2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by The stalk of a presheaf at a point.

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

The Stacks Project, Sheaves on Spaces, Section 11 (standard reference, not scraped)