ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge 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.

A generic point and a distinct specialization cannot be separated in the Zariski topology

Example

The Zariski topology on $Spec (Z)$ is not Hausdorff.

Facts & Assumptions

Given: The points $(0)$ and $(2)$ of $Spec (Z)$ .

[L1]

In $Spec (Z)$ , the point $(0)$ is generic and $(2)$ is a specialization of it (The spectrum of the integers has one generic point, closed points (p), and basic opens D(n)).

[L2]

Specialisation in a spectrum is reverse inclusion (Specialisation in a prime spectrum is reverse inclusion).

[A1]

If $y$ is a specialization of $x$ , then every open neighborhood of $y$ contains $x$ .

Verification

technique · direct

1.1

By [L1], $(2)$ is a specialization of $(0)$ . Equivalently, [L2] records the containment $(0) \subseteq (2)$ .

L1L2

2.1

By [A1], every open neighborhood of $(2)$ contains $(0)$ . Hence no open neighborhood of $(0)$ can be disjoint from an open neighborhood of $(2)$ . So the two distinct points cannot be separated by disjoint open sets.

A1step 1.1

3.1

Therefore $Spec (Z)$ is not Hausdorff.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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, §14 (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 (standard reference, not scraped)