DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Noetherian topological spaces via ACC on opens or DCC on closed subsets

The space $X$ is Noetherian when every ascending chain $U_{0} \subseteq U_{1} \subseteq U_{2} \subseteq \dots$ of open subsets stabilizes.

Equivalently, $X$ is Noetherian when every descending chain $F_{0} \supseteq F_{1} \supseteq F_{2} \supseteq \dots$ of closed subsets stabilizes.

Depends on