DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

The category of open subsets of a topological space

Definition

Let $X$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

The open-set category $Open (X)$ has:

objects: the open subsets $U \subseteq X$ ;
morphisms: for opens $V, U$ , a unique morphism $V \to U$ when $V \subseteq U$ , and no morphism otherwise.

Composition is forced by transitivity of inclusion, and identity morphisms are the inclusions $U \subseteq U$ . Thus $Open (X)$ is a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).

The arrows point in the same direction as inclusion. Therefore a contravariant functor on $Open (X)$ sends an inclusion $V \subseteq U$ to a restriction map from the larger open set to the smaller one.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Sheaves on Spaces, Section 2