CounterexampleConstruction: AI-generatedVerification: AI-generatedaudited 2026-09-12

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.

The Zariski topology on the affine line over an infinite field is not Hausdorff

Statement refuted

Over an infinite field, the Zariski topology on $A_{k}^{1}$ is Hausdorff.

Example

Let $a \neq b$ be two points of $A_{k}^{1}$ . By On the affine line, the classical Zariski topology is cofinite, every nonempty open subset of $A_{k}^{1}$ is the complement of a finite set. If $U$ and $V$ are nonempty open neighborhoods of $a$ and $b$ , then $U \cap V = A_{k}^{1} ∖ (F \cup G)$ for finite sets $F$ and $G$ . Because $k$ is infinite, $F \cup G$ is still a proper subset of $k$ , so $U \cap V \neq \emptyset$ .

Thus no two distinct points admit disjoint neighborhoods. The refuted statement is therefore false: the classical Zariski topology on $A_{k}^{1}$ is not Hausdorff.

Depends on

On the affine line, the classical Zariski topology is cofinite

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Donu Arapura, Notes on Basic Algebraic Geometry, cofinite-topology discussion in §1.3 (standard reference, not scraped)