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

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Equivalent analytic normal forms and Borel maps

Statement

In ZFC, for $A$ in a Polish $X$ , these conditions are equivalent: $A$ is analytic in the closed-projection convention; $A$ is empty or a continuous image of $N$ ; $A$ is a continuous image of a Borel subset of a Polish space; $A$ is the projection of a Borel subset of $Y \times X$ for some Polish $Y$ . Analytic sets are preserved by Borel measurable images and inverse images between Polish spaces; coanalytic sets are preserved by such inverse images. A map is Borel measurable if its open preimages are Borel.

Facts & Assumptions

[F1]

Closed subspaces, products, and Baire parametrization gives Polish closed products and Baire parametrization of nonempty Polish spaces.

[F2]

Analytic countable operations and inclusion of Borel sets gives Borel inclusion, intersections, and continuous inverse images for analytic sets.

[F3]

Analytic and coanalytic sets by closed projection fixes the analytic and coanalytic conventions.

[F4]

The countable Borel hierarchy and its limit convention defines the Borel sigma-algebra as the least open-containing sigma-algebra.

[A1]

Assume The Axiom of Choice.

Proof

Given: The Polish spaces and ZFC assumptions in the statement.

1.1

If $A$ is a nonempty closed projection with witness $F \subseteq X \times N$ , F1 makes $F$ Polish and gives a continuous surjection $h : N \to F$ . Projection composed with $h$ is continuous onto $A$ . Conversely if $f : N \to X$ is continuous, its reversed graph ${(f (y), y) : y \in N}$ is closed. Indeed if $x \neq f (y)$ , disjoint metric neighbourhoods of these two points and continuity at $y$ give a product neighbourhood of $(x, y)$ missing the graph. The graph projects to $f [N]$ , giving analyticity by F3. Empty $A$ has the empty closed witness.

F1F3

1.2

Let $f : Y \to X$ be Borel measurable. If $X$ is empty then $Y$ is empty and its graph is empty; otherwise enumerate, for every $n$ , all basic opens $V_{n j}$ in a countable metric basis of $X$ having diameter less than $2^{- n}$ . They cover $X$ , by dense-centre rational balls. Then

$graph (f) = ⋂_{n} ⋃_{j} (f^{- 1} [V_{n j}] \times \overline{V_{n j}}) .$

The forward inclusion uses a basis member containing $f (y)$ . For the reverse, membership on the right gives for every $n$ a set containing $f (y)$ whose closure contains $x$ , and thus $d (f (y), x) \leq 2^{- n}$ ; hence $x = f (y)$ . Each rectangle is Borel: coordinate projections have Borel preimages of Borel sets, because the family of sets with Borel preimage is a sigma-algebra containing opens, by continuity and F4. Intersect their two coordinate preimages and use F4's countable operations. Hence the graph is Borel. [F4]

2.1

A continuous image of any analytic set $B$ is analytic: for nonempty $B$ compose its parametrization from step 1.1 with the given continuous map; for empty $B$ use the empty witness. Borel subsets are analytic by F2, so this proves that the third condition implies the first, including maps defined only on the Borel subset (the parametrization is continuous into that subspace). The second implies the third by taking the domain $N$ or the empty Borel domain. The fourth implies the first by projecting a Borel, hence analytic, subset of the Polish product supplied by F1. The first implies the fourth using its closed witness, with coordinates reversed. Thus all four conditions are equivalent.

F1F2step 1.1

3.1

For analytic $B \subseteq Y$ , its cylinder $B \times X$ is analytic by F2's continuous inverse-image closure. The graph is analytic by F2 and step 1.2, so their intersection is analytic by F2. Its continuous projection onto $X$ is $f [B]$ , analytic by step 2.1. For analytic $A \subseteq X$ , instead intersect the graph with $Y \times A$ and project to $Y$ , obtaining $f^{- 1} [A]$ by the same argument. All products are Polish by F1. Finally if $A$ is coanalytic, $X ∖ A$ is analytic by F3, and $Y ∖ f^{- 1} [A] = f^{- 1} [X ∖ A]$ is analytic by what was just proved. This gives the coanalytic inverse-image claim. All invoked ZFC suppliers are licensed by A1. QED.

F1F2F3A1step 2.1step 1.2

Depends on

Used by

Analytic boundedness for well-founded trees Theorem
Borel separation of disjoint analytic sets Theorem
Closed Souslin schemes characterize analytic sets Theorem
Uncountable analytic sets contain compact Cantor copies Theorem

Dependency tree · two levels

15 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

Definition 4.1, Lemma 4.2 p34 and Lemma 4.5(ii–iii) pp34–35; graph argument supplied locally instead of importing Theorem 2.27 (standard reference, not scraped)