TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

Assuming countable choice, every second countable space is separable

Statement

Assuming $\mathrm{AC}_\omega$ , every second countable space is separable.

Facts & Assumptions

Given: A countable basis $\mathcal B$ for $X$ .

[A1]

Countable choice selects one element from every nonempty member of a countable family (The Axiom of Countable Choice ( $\mathrm{AC}_\omega$ )).

Proof

technique · constructive

1.1

Apply [A1] to the nonempty members of $\mathcal B$ , and let $D$ be the selected points.

A1construct

2.1

The set $D$ is countable and meets every nonempty basic open set, hence every nonempty open set, so it is dense.

step 1.1

3.1

Therefore $X$ is separable.

step 2.1discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 10 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

UCR General Topology Notes (standard reference, not scraped)