CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 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.

The constant presheaf need not be a sheaf on a disconnected open set

Statement refuted

Fix a set $A$ with distinct elements $a \neq b$ . The constant presheaf ${\underline{A}}_{p r e}$ on a space $X$ , defined by ${\underline{A}}_{p r e} (U) = A$ for every open set $U$ and identity restriction maps, is always a sheaf.

Facts & Assumptions

Given: A space $X$ containing a disconnected open set $U = U_{1} ⊔ U_{2}$ with $U_{1}, U_{2} \neq \emptyset$ , and a set $A$ with $a \neq b$ .

[F1]

Identity restriction maps define a presheaf on $X$ (A presheaf on a topological space).

[F2]

The sheafification $a F$ of a presheaf is defined by the double plus construction (Sheafification of a presheaf).

[L1]

Locally constant $A$ -valued functions form a sheaf (Locally constant functions form a sheaf and have constant stalks).

Counterexample

technique · direct

1.1

By [F1], ${\underline{A}}_{p r e}$ is a presheaf. On the disjoint cover $U = U_{1} \cup U_{2}$ , choose the local sections $s_{1} : = a \in {\underline{A}}_{p r e} (U_{1})$ and $s_{2} : = b \in {\underline{A}}_{p r e} (U_{2})$ . Because $U_{1} \cap U_{2} = \emptyset$ , there is no overlap condition to check, so the pair is compatible.

F1givenconstruct

2.1

A glued section over $U$ would have to be an element $s \in A$ whose restriction to $U_{1}$ is $a$ and to $U_{2}$ is $b$ . But every restriction map is the identity, so this would force $s = a = b$ , contradicting the choice $a \neq b$ . Therefore ${\underline{A}}_{p r e}$ is not a sheaf.

step 1.1given

3.1

The sheafification of ${\underline{A}}_{p r e}$ records exactly the data obtained by gluing constant local sections on an open cover, which is the same as an $A$ -valued locally constant function. By [L1], that sheaf is ${\underline{A}}_{l o c}$ , so [F2] identifies the locally constant sheaf as the sheafification of the constant presheaf.

F2L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, Definition 3.2 (standard reference, not scraped)
The Stacks Project, Sheaves on Spaces, Definition 7.4 (standard reference, not scraped)