CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Borel sets are exactly analytic and coanalytic sets

Statement

In ZFC, a subset of a Polish space is Borel if and only if it is analytic and coanalytic.

Facts & Assumptions

[F1]

Analytic countable operations and inclusion of Borel sets makes every Borel set analytic and coanalytic.

[F2]

Borel separation of disjoint analytic sets separates disjoint analytic sets by a Borel set.

[A1]

Assume The Axiom of Choice.

Proof

Given: A subset $A$ of Polish $X$ , under ZFC.

1.1

If $A$ is Borel, F1, whose ZFC hypothesis is supplied by A1, says exactly that $A$ and its complement are analytic. This is analyticity and coanalyticity.

F1A1

2.1

Conversely if $A$ is analytic and coanalytic, both $A$ and $X ∖ A$ are analytic, and they are disjoint. By F2 and A1 take Borel $C$ with $A \subseteq C$ and $C \cap (X ∖ A) = \emptyset$ . The second relation gives $C \subseteq A$ , so $A = C$ is Borel. This includes $A = \emptyset$ and $A = X$ by the same inclusions. QED.

F2A1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Corollary 4.14, printed p37 (standard reference, not scraped)