TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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 Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra

Statement

Let $Y$ be a subspace of a topological space $X$ . Then

$B (Y) = B (X) ∣_{Y} .$

Facts & Assumptions

Given: A topological space $X$ and a subset $Y \subseteq X$ with its subspace topology.

[L1]

The subspace topology on $Y$ consists exactly of the traces $U \cap Y$ of open sets $U$ of $X$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

[L2]

Generating a sigma-algebra commutes with taking traces (Generating a sigma-algebra commutes with taking traces).

[L3]

The Borel sigma-algebra of a topological space is generated by its open sets (The Borel sigma-algebra of a topological space).

Proof

technique · direct

1.1

By [L1], the family generating $B (Y)$ is precisely the trace on $Y$ of the family generating $B (X)$ .

L1L3

2.1

Applying [L2] to the family of open subsets of $X$ gives $B (Y) = B (X) ∣_{Y}$ .

step 1.1L2L3∎

Depends on

Used by

The Borel sigma-algebra of the Cantor set is the trace of the real Borel sigma-algebra Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 8 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

T. Tao, An Introduction to Measure Theory, Exercise 1.4.12 (standard reference, not scraped)