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.
Every separable space satisfies the countable chain condition
Statement
Every separable space is ccc.
Facts & Assumptions
Given: A countable dense set and a pairwise-disjoint family of nonempty open sets.
A nonempty countable set can be enumerated by natural numbers (A nonempty set is at most countable iff it is a surjective image of ).
Proof
If , it is already at most countable. Otherwise is nonempty, so the dense set is nonempty; enumerate and assign to each the first enumerated point of , which is nonempty by density.
Disjointness makes this assignment injective into a countable set.
Hence is countable and is ccc.
Depends on
- Separability: the existence of an at most countable dense subset
- The countable chain condition: every pairwise-disjoint family of nonempty open sets is at most countable
- Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets
- A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$
Used by
Dependency tree · two levels
16 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)
- D. H. Fremlin, Measure Theory, Chapter 5A (standard reference, not scraped)
- Countable chain condition (Wikipedia) (standard reference, not scraped)