Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31
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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.

Stone's theorem, under choice: every metric space is paracompact

Statement

Assume the Axiom of Choice. Every metric space is paracompact.

Facts & Assumptions

Given: The Axiom of Choice, a metric space X, and an arbitrary open cover U of its metric topology.

[L1]

Under choice, every metric open cover has a point-finite open refinement, and Ornstein's second construction turns that point-finite cover into a locally finite open refinement (Under choice, every open cover of a metric space has a point-finite open refinement, Under choice, Ornstein's second construction turns a point-finite metric open cover into a locally finite open refinement).

[F1]

Paracompactness means that every open cover has such a refinement (Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word).

Proof

technique · direct
1.1L1

Apply [L1] to the arbitrary cover U.

2.1F1step 1.1∎

The resulting locally finite open refinement is exactly the condition in [F1], so X is paracompact.

Remarks

The theorem is proved here with the Axiom of Choice as a sufficient hypothesis. No assertion is made that this is its exact set-theoretic strength.

Depends on

Dependency tree · 2 levels

Sources