LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Every discrete family is locally finite, so every $\sigma$ -discrete basis is $\sigma$ -locally finite

Statement

Every discrete family is locally finite. Consequently every $\sigma$ -discrete basis is $\sigma$ -locally finite.

Facts & Assumptions

Given: A discrete family $\mathcal D$ in a space $X$ .

[L1]

A family is locally finite when every point has a neighbourhood meeting only finitely many of its members (Refinements, locally finite families, point-finite families, and star refinements).

Proof

technique · direct

1.1

For each $x\in X$ , discreteness supplies a neighbourhood meeting at most one member of $\mathcal D$ . That is a finite number, so [L1] makes $\mathcal D$ locally finite.

2.1

Applying step 1.1 separately to every discrete layer in a $\sigma$ -discrete decomposition leaves the same countable decomposition and gives a $\sigma$ -locally-finite basis.

step 1.1∎

Depends on

Used by

Under choice, a space is metrizable if and only if it is regular, T₁, and has a σ-discrete basis Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 9 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)

Every discrete family is locally finite, so every σ\sigmaσ-discrete basis is σ\sigmaσ-locally finite