Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17
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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 continuous map has Borel preimages of Borel sets

Statement

If f:X→Y is continuous between topological spaces, then f−1[B]∈B(X) for every B∈B(Y).

Facts & Assumptions

Given: Topological spaces X,Y and a continuous map f:X→Y.

[L1]

Continuity implies that f−1[V] is open in X for every open V⊆Y (For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(A‾)⊆f(A)‾).

[L2]

The Borel sigma-algebra is the smallest sigma-algebra containing the open sets (The Borel sigma-algebra of a topological space, Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal).

Proof

technique · direct
1.1

Let C:={B⊆Y:f−1[B]∈B(X)}. Preimages preserve complements and countable unions, so C is a sigma-algebra on Y.

L2algebra
1.2

Every open V⊆Y belongs to C, because [L1] makes f−1[V] open and therefore Borel in X.

L1L2
2.1

Minimality in [L2] gives B(Y)⊆C, which is the stated conclusion.

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources