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 Lindelöf

Statement

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

Facts & Assumptions

Given: A countable basis $\mathcal B$ and an open cover $\mathcal U$ of $X$ .

[A1]

Countable choice selects from the nonempty families indexed by the eligible basis members (The Axiom of Countable Choice ( $\mathrm{AC}_\omega$ )).

Proof

technique · constructive

1.1

For each $B\in\mathcal B$ that lies in some $U\in\mathcal U$ , use [A1] to select one such $U_B$ .

A1construct

2.1

The selected family is countable and covers $X$ : a point lies in a cover member, and a basis member containing it lies inside that member.

step 1.1

3.1

Thus every open cover has a countable subcover, so $X$ is Lindelöf.

step 2.1discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 16 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

nLab: second-countable spaces are Lindelöf (standard reference, not scraped)
UCR General Topology Notes (standard reference, not scraped)