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.
Assuming countable choice, every second countable space is separable
Statement
Assuming , every second countable space is separable.
Facts & Assumptions
Given: A countable basis for .
Countable choice selects one element from every nonempty member of a countable family (The Axiom of Countable Choice ()).
Proof
Apply [A1] to the nonempty members of , and let be the selected points.
The set is countable and meets every nonempty basic open set, hence every nonempty open set, so it is dense.
Therefore is separable.
Depends on
Used by
- Assuming choice and countable choice, refuted: arbitrary products of second countable spaces are second countable False statement
- Hausdorff and locally Euclidean do not by themselves make a manifold False statement
- Implication, preservation, counterexample, and choice ledger for the countability axioms Remark
- Under Dependent Choice, a subspace of a Polish space is Polish exactly when it is G_δ Theorem
Dependency tree · two levels
12 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)