LemmaStatement: AI-generatedProof: AI-generated

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.

Open subsets containing generic points are componentwise dense

PDF p. 10: Dense smooth locus in Lemma 5.1

Statement

Let $U$ be an open subset of a topological space $X$ . If an irreducible component $C$ has generic point $η \in U$ , then $U \cap C$ is a dense open subset of $C$ .

The generic point condition means $\overline{{η}} = C$ ; density means that the closure in $C$ is all of $C$ .

Facts & Assumptions

Given: An open $U \subseteq X$ and a component $C$ with generic point $η \in U$ .

Goal: Prove that $U \cap C$ is open and dense in $C$ .

Proof

1.1given

The intersection $U \cap C$ is open in the subspace topology on $C$ and contains $η$ .

2.1step 1.1given∎

Its closure in $C$ contains the closure of ${η}$ in $C$ , which is $C$ . Hence it is dense in $C$ .

Used by · 1

Smooth density and normal codimension over a perfect field