TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02‡ rests on unproved material

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.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Under choice, a space is metrizable if and only if it is regular, $T_{1}$ , and has a $σ$ -locally-finite basis

Statement

Assume the Axiom of Choice. A space is metrizable if and only if it is regular, $T_{1}$ , and has a $σ$ -locally-finite basis. Here regularity and $T_{1}$ are separate hypotheses.

Facts & Assumptions

Given: The Axiom of Choice and a topological space $X$ .

[L1]

Under choice every metric space has a $σ$ -locally-finite basis (Under choice, every metric space has a $σ$ -locally-finite basis).

‡ [L2]

Recorded external fallback; not proved here. A regular $T_{1}$ space with such a basis has a compatible normal sequence (Under choice, a regular $T_{1}$ space with a $σ$ -locally-finite basis has a compatible normal sequence ‡); a $T_{1}$ space with such a sequence is metrizable (A $T_{1}$ space with a compatible normal sequence of open covers is metrizable).

Proof

technique · cases

1.1

If $X$ is metrizable, it is regular and $T_{1}$ , and [L1] gives the required basis.

assume-case forwardL1

1.2

If $X$ is regular, $T_{1}$ , and has a $σ$ -locally-finite basis, [L2] first gives a compatible normal sequence and then a compatible metric.

assume-case reverse‡ L2

2.1

The two cases prove the two directions of the equivalence.

step 1.1step 1.2cases-exhaustive∎

Depends on

Used by

Dependency tree · two levels

34 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

Encyclopedia of Mathematics, Metrizable space (standard reference, not scraped)

Under choice, a space is metrizable if and only if it is regular, T1T1​, and has a σσ-locally-finite basis