TheoremStatement: AI-adaptedProof: AI-generated

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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.
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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 (The Axiom of Choice) and a topological space $X$ .

[L1]

Under choice every metric space has a $σ$ -discrete open basis, by the closed-core and open-neighborhood construction of Under choice, metric spaces have sigma-discrete open bases.

[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

1.1assume-case forwardL1

If $X$ is metrizable, [L1] gives a $σ$ -discrete basis; metric spaces are regular and $T_{1}$ : singleton complements are open, and for a point $x$ outside a closed set $F$ , a ball $B (x, r)$ missing $F$ has the closure of $B (x, r / 2)$ still missing $F$ , by the triangle inequality (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

1.2assume-case reverseL2

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.

2.1step 1.1step 1.2cases-exhaustive∎

The cases establish both directions.

Depends on

Used by

A sigma-cellular base metrizes a normal Moore space Lemma

Dependency tree · two levels

27 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