LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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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Locally finite supports have locally finite cozero sets

Statement

Let $(f_{i})_{i \in I}$ be a family of real-valued functions on a topological space $X$ . If the family of supports $(supp (f_{i}))_{i \in I}$ is locally finite, then the family of cozero sets $({x : f_{i} (x) \neq 0})_{i \in I}$ is locally finite.

Facts & Assumptions

Given: A family $(f_{i})_{i \in I}$ of real-valued functions on $X$ .

[F1]

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

[A1]

For each $i$ , the cozero set of $f_{i}$ is contained in $supp (f_{i})$ .

Proof

technique · direct

1.1

Fix $x \in X$ ; by [F1], there is a neighbourhood $N$ of $x$ meeting only finitely many supports $supp (f_{i})$ .

F1given

2.1

If $N$ meets the cozero set of $f_{i}$ , then it meets $supp (f_{i})$ by [A1], so only finitely many cozero sets can meet $N$ .

A1step 1.1

3.1

Since $x$ was arbitrary, the cozero family is locally finite by [F1].

F1step 2.1∎

Depends on

Refinements, locally finite families, point-finite families, and star refinements

Used by

Dependency tree · two levels

4 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)