TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Assuming countable choice, every second countable space is Lindelöf

Assuming ${A C}_{ω}$ , every second countable space is Lindelöf.

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

[A1]

Countable choice selects from the nonempty families indexed by the eligible basis members (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

technique · constructive

1.1A1construct

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

2.1step 1.1

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.

3.1step 2.1discharge-construct∎

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

Depends on