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 $σ$ -discrete basis

Statement

Assume the Axiom of Choice. A space is metrizable if and only if it is regular, $T_{1}$ , and has a $σ$ -discrete basis.

Facts & Assumptions

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

‡ [L1]

Recorded external fallback; not proved here. Under choice every metric space has a $σ$ -discrete basis (Under choice, every metric space has a $σ$ -discrete basis ‡).

[L2]

A discrete family is locally finite, and Nagata–Smirnov metrizes every regular $T_{1}$ space with a $σ$ -locally-finite basis (Every discrete family is locally finite, so every $σ$ -discrete basis is $σ$ -locally finite, Under choice, a space is metrizable if and only if it is regular, $T_{1}$ , and has a $σ$ -locally-finite basis).

Proof

technique · cases

1.1

If $X$ is metrizable, [L1] gives a $σ$ -discrete basis; metric spaces are regular and $T_{1}$ .

assume-case forward‡ L1

1.2

If $X$ is regular, $T_{1}$ , and has a $σ$ -discrete basis, [L2] converts that basis to a $σ$ -locally-finite one and then gives a metric.

assume-case reverseL2

2.1

The cases establish both directions.

step 1.1step 1.2cases-exhaustive∎

Depends on

Used by

A sigma-cellular base metrizes a normal Moore space Lemma

Dependency tree · two levels

16 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

R. H. Bing, Metrization of Topological Spaces (standard reference, not scraped)

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