Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02
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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.

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

Statement

Every discrete family is locally finite. Consequently every σ-discrete basis is σ-locally finite.

Facts & Assumptions

Given: A discrete family 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∈X, discreteness supplies a neighbourhood meeting at most one member of D. That is a finite number, so [L1] makes D locally finite.

L1
2.1

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

step 1.1∎

Depends on

Used by

Dependency tree · two levels

7 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