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.

Irreducible topological spaces and irreducible subsets in the subspace topology

Definition

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

The space $X$ is irreducible when $X \neq \emptyset$ and whenever $X = F_{1} \cup F_{2}$ with $F_{1}, F_{2} \subseteq X$ closed, one has $X = F_{1}$ or $X = F_{2}$ .

If $A \subseteq X$ , then $A$ is an irreducible subset of $X$ when the subspace $(A, T_{A})$ is irreducible, where $T_{A}$ is the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

Depends on

Used by

Dependency tree · two levels

6 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

J. S. Milne, A Primer of Commutative Algebra, v4.03, Definition 14.5 (standard reference, not scraped)
The Stacks Project, Definition 5.8.1 (standard reference, not scraped)