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 proved Stone theorem and its effective strengthening
Statement
AC proves that every metric space is paracompact (Stone's theorem, under choice: every metric space is paracompact). The stronger assertion that every open cover of every discrete metrizable space has a point-finite refinement equipped with a refinement map implies AC (Effective metacompactness for discrete metric spaces implies AC).
Remarks
-
The ordinary theorem. The AC hypothesis in the local Stone theorem is sufficient for its proof. The theorem supplies a locally finite open refinement of every open cover; it asserts no global refinement operator.
-
The effective strengthening. The refinement map is part of the hypothesis in Effective metacompactness for discrete metric spaces implies AC, and the local proof recovers a choice function from that map. This stronger assertion is not identified with the ordinary existential Stone theorem.
-
The separate DC row. Relative consistency of DC with failure of Stone's theorem is the existing supplier for the conditional comparison with DC. The present amendment removes the unsupported BPI transfer comparison and makes no claim about the current research status of the exact ordinary theorem.
Depends on
Used by
Dependency tree · two levels
23 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
- C. Good, I. J. Tree, and W. S. Watson, On Stone's theorem and the axiom of choice (standard reference, not scraped)
- Samuel Corson, The Independence of Stone's Theorem from the Boolean Prime Ideal Theorem (standard reference, not scraped)