LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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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.
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Irreducibility is equivalent to the nonempty-open intersection criterion

Statement

For a nonempty topological space $X$ , irreducibility is equivalent to the intersection of every two nonempty open subsets being nonempty. Every nonempty open subspace of an irreducible space is itself irreducible and dense. This applies to the classical Zariski spaces.

Facts & Assumptions

Given: A nonempty topological space $X$ . For the inheritance assertions assume $X$ irreducible and let $U \subseteq X$ be nonempty open.

[F1]

Irreducibility excludes a union of two proper closed subsets (A classical affine variety).

Proof

technique · direct

1.1

Two disjoint nonempty opens $U, V$ give the proper closed cover $X = (X ∖ U) \cup (X ∖ V)$ . Conversely a proper closed cover $X = C \cup D$ gives disjoint nonempty opens $X ∖ C, X ∖ D$ . Taking complements proves both directions of the criterion.

F1given

2.1

If $U$ is nonempty open in irreducible $X$ , each nonempty open $W$ of $X$ meets $U$ by step 1.1. Thus no proper closed subset of $X$ contains $U$ , which says $\overline{U} = X$ . If $V_{1}, V_{2}$ are nonempty opens of $U$ , they are opens of $X$ because $U$ is open, so they intersect. The criterion makes $U$ irreducible.

step 1.1given∎

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, §2h p. 45. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

A classical affine variety

Used by

Dependency tree · two levels

3 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, Algebraic Geometry v6.10, §2h p. 45 (standard reference, not scraped)