DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The countable chain condition: every pairwise-disjoint family of nonempty open sets is at most countable

Definition

A topological space $X$ satisfies the countable chain condition (ccc) if every family $\mathcal U$ of nonempty open subsets of $X$ with $U\cap V=\varnothing$ whenever $U,V\in\mathcal U$ are distinct is at most countable (Finite, countably infinite, countable, uncountable).

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 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)
Countable chain condition (Wikipedia) (standard reference, not scraped)
D. H. Fremlin, Measure Theory, Chapter 5A (standard reference, not scraped)