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

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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

1.1F1given

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

2.1A1step 1.1

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 $N$ .

3.1F1step 2.1∎

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

Depends on

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

Locally finite supports have locally finite cozero sets

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources