LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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Dimension of a finite closed union

Statement

If a Noetherian space $T$ is a finite union of closed subsets $T_{1}, \dots, T_{m}$ , then $\dim T = \max_{i} \dim T_{i}$ . For $m = 0$ both sides are $- \infty$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

For a Noetherian topological space $T$ , define $\dim T$ as the supremum of the lengths $s$ of strict chains $Z_{0} ⊊ \dots ⊊ Z_{s}$ of nonempty irreducible closed subsets of $T$ . Thus a one-member chain has length zero. Set $\dim \emptyset = - \infty$ , and allow $\dim T = + \infty$ . The supremum of an empty family of dimensions is $- \infty$ . (Chain dimension and the empty-space convention).

Proof

1.1

If $m = 0$ or $T = \emptyset$ , there are no nonempty irreducible closed subsets and the conventions give the equality. Otherwise every chain in a closed $T_{i}$ is also a chain in $T$ , giving $\dim T \geq \dim T_{i}$ .

2.1

For a chain $Z_{0} ⊊ \dots ⊊ Z_{s}$ in $T$ , write $Z_{s} = ⋃_{i} (Z_{s} \cap T_{i})$ . Irreducibility, applied repeatedly to this finite closed cover, forces $Z_{s} \subseteq T_{i}$ for some $i$ . The entire chain lies in that $T_{i}$ . Taking suprema gives the reverse bound, including unbounded chain lengths.

F1step 1.1∎

Depends on

Chain dimension and the empty-space convention

Used by

Closed families with irreducible equal-dimensional fibres Corollary
Image dimension and the generic fibre formula Corollary
Several homogeneous equations in projective space Corollary
Global and local dimension of classical varieties Definition
Plane curves meet; common components change the dimension Example
The coordinate cross has two one-dimensional components Example
The family xy=t has constant dimension and a reducible special fibre Example
A nonempty projective cone raises dimension by one Lemma
Dimension is detected by avoiding linear subspaces Lemma
Dimensions add under products Theorem

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

Milne §2m 2.49 p.54: maximum over irreducible components (standard reference, not scraped)