CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02‡ rests on unproved material (inherited)

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, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Under choice, a regular T₁ space with a σ-locally-finite basis has a compatible normal sequence. Each is recorded with a citation to the literature and 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, every regular $T_1$ second-countable space is metrizable

Statement

Assume the Axiom of Choice. Every regular $T_1$ second-countable space is metrizable.

Facts & Assumptions

Given: A regular $T_1$ space with a countable basis $(B_n)_{n\in\mathbb N}$ , and the Axiom of Choice.

[L1]

Nagata–Smirnov metrizes a regular $T_1$ space with a $\sigma$ -locally-finite basis (Under choice, a space is metrizable if and only if it is regular, $T_1$ , and has a $\sigma$ -locally-finite basis).

Proof

technique · direct

1.1

Each singleton family $\{B_n\}$ is locally finite, including when $B_n=\varnothing$ . Hence the displayed basis is $\sigma$ -locally finite.

given

2.1

Apply [L1] to step 1.1 and the given regular and $T_1$ hypotheses.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

Under choice, every regular T1T_1T1​ second-countable space is metrizable