DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-21

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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

Used by

Dependency tree · two levels

21 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

V. Guillemin, MIT 18.101 Analysis II Lecture Notes, §§3.7–3.8 (standard reference, not scraped)

Compact Jordan exhaustions of open subsets of RnRn