PropositionStatement: 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.

Every separable space satisfies the countable chain condition

Statement

Every separable space is ccc.

Facts & Assumptions

Given: A countable dense set $D$ and a pairwise-disjoint family $\mathcal U$ of nonempty open sets.

[L1]

A nonempty countable set can be enumerated by natural numbers (A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$ ).

Proof

technique · direct

1.1

If $\mathcal U=\varnothing$ , it is already at most countable. Otherwise $X$ is nonempty, so the dense set $D$ is nonempty; enumerate $D$ and assign to each $U\in\mathcal U$ the first enumerated point of $D\cap U$ , which is nonempty by density.

givenL1

2.1

Disjointness makes this assignment injective into a countable set.

step 1.1

3.1

Hence $\mathcal U$ is countable and $X$ is ccc.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 14 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)
D. H. Fremlin, Measure Theory, Chapter 5A (standard reference, not scraped)
Countable chain condition (Wikipedia) (standard reference, not scraped)