DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Compact support, $C_{c} (X)$ , and $C_{0} (X)$

Definition

Let $X$ be locally compact Hausdorff and let $f : X \to F$ be continuous, where $F$ is $R$ or $C$ . Its support is $supp f : = \overline{{x : f (x) \neq 0}}$ . Put $C_{c} (X; F) = {f \in C (X; F) : supp f is compact}$ . Also $C_{0} (X; F)$ consists of those $f$ for which, for every $ϵ > 0$ , ${x : ∣ f (x) ∣ \geq ϵ}$ is compact. We write $C_{c} (X)$ for the real space until the bounded complex $C_{0}$ theorem is invoked; $C_{0} (X)$ later means the complex space when its scalar field matters.

Depends on

Dependency tree · 2 levels

Sources

Donald L. Cohn, Measure Theory, 2nd ed., §7.1

Compact support, Cc(X)Cc​(X), and C0(X)C0​(X)

Definition

Depends on

Dependency tree · 2 levels

Sources

Compact support, $C_{c} (X)$ , and $C_{0} (X)$