ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Continuous real-valued functions form a sheaf

Example

For each open set $U \subseteq X$ , let $C^{0} (U, R) : = {f : U \to R continuous} .$ With the usual restriction of functions, this is a sheaf of sets on $X$ .

Facts & Assumptions

Given: An open cover $U = ⋃_{i \in I} U_{i}$ of an open set $U$ .

[L1]

A sheaf is exactly a presheaf with locality and unique gluing on every open cover (A sheaf on a topological space).

Verification

technique · direct

1.1

Restriction of a continuous function is continuous, so $U \mapsto C^{0} (U, R)$ is a presheaf. If two continuous functions on $U$ agree on every $U_{i}$ , then they agree pointwise on all of $U$ because the $U_{i}$ cover $U$ .

given

2.1

Let $f_{i} \in C^{0} (U_{i}, R)$ be compatible on overlaps. Define $f : U \to R$ by $f (x) = f_{i} (x)$ for any $i$ with $x \in U_{i}$ . Compatibility makes this well defined, and continuity is local on the open cover because $f ∣_{U_{i}} = f_{i}$ is continuous for each $i$ . Thus [L1] holds.

L1givenconstruct∎

Depends on

A sheaf on a topological space

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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, Example 7.3 (standard reference, not scraped)