Alphabeta Math
RemarkRemark: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)‡ sources checked 2026-08-01‡ not proved here
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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Under choice, every metric space has a σ\sigma-discrete basis

Statement

Assume the Axiom of Choice. Every metric space has a σ\sigma-discrete open basis.

Not proved in this library. This is a source-backed fallback rather than a local proof. The discarded construction assigns every point to a first centre and then asserts that the resulting cells are open. They need not be: in R\mathbb R, with integer centres at scale 11 ordered 0,1,−1,2,…0,1,-1,2,\ldots, the cell assigned to 11 is [1,2)[1,2). Intersecting it with a positive-distance condition does not repair that failure.

Why it remains visible. Bing's standard necessity direction needs a real discrete-open refinement argument. The external-dependency marker is retained on this result and its consequences so that none of them is mistaken for a locally established theorem.

Used by

Dependency tree · next 3 levels

Nothing. This result depends on no other item in the library.

Sources