DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-21

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 Jordan exhaustions of open subsets of $R^{n}$

Definition

Let $n \geq 1$ and let $D \subseteq R^{n}$ be open. A compact Jordan exhaustion of $D$ is a sequence $(K_{j})_{j \in N}$ such that:

every $K_{j} \subseteq D$ is compact (Open cover, subcover, compact metric space, and compact subset of a metric space) and Jordan measurable (Jordan inner and outer content and Jordan measurable bounded sets in $R^{m}$ );
$K_{j} \subseteq int K_{j + 1}$ for every $j$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space);
$D = ⋃_{j \in N} K_{j}$ .

These clauses imply compact cofinality: every compact $C \subseteq D$ is contained in some $K_{j}$ . Indeed, the open sets $int K_{j + 1}$ cover $C$ ; compactness gives a finite subcover, and nesting places all of $C$ in the member with largest index. For $D = \emptyset$ , the constant sequence $K_{j} = \emptyset$ is an exhaustion.

Depends on

Dependency tree · 2 levels

Sources

V. Guillemin, MIT 18.101 Analysis II Lecture Notes, §§3.7–3.8

Compact Jordan exhaustions of open subsets of RnRn

Definition

Depends on

Dependency tree · 2 levels

Sources

Compact Jordan exhaustions of open subsets of $R^{n}$