Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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⊆X;
  • morphisms: for opens V,U, a unique morphism V→U when V⊆U, and no morphism otherwise.

Composition is forced by transitivity of inclusion, and identity morphisms are the inclusions U⊆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⊆U to a restriction map from the larger open set to the smaller one.

Depends on

Used by

Dependency tree · two levels

5 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