Alphabeta MathHow 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 satisfies the countable chain condition (ccc) if every family of nonempty open subsets of with whenever are distinct is at most countable (Finite, countably infinite, countable, uncountable).
Depends on
Used by
- Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ₀ threshold Corollary
- Every separable space satisfies the countable chain condition Proposition
- Under choice, every Cantor cube 2^I satisfies ccc Theorem
- Under MA(aleph₁), arbitrary products of ccc spaces are ccc Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)