Compact exhaustion of an open Euclidean set

A compact exhaustion of an open set $U\subseteq\mathbb R^n$ is a sequence of compact sets $K_j\subset U$ with

K_j\subseteq\operatorname{int}K_{j+1},\qquad U=\bigcup_j\operatorname{int}K_j.

Every compact subset of $U$ is then contained in some $K_j$ : take a finite subcover from the increasing cover by interiors.

Explicit construction

If $U\ne\mathbb R^n$ , one may take

K_j=\{x\in\mathbb R^n:|x|\le j,\quad \operatorname{dist}(x,\mathbb R^n\setminus U)\ge1/j\},\qquad j\ge1.

These are compact by the Heine–Borel theorem and lie inside $U$ . The strict improvement of both inequalities at the next index gives the interior inclusion. For $U=\mathbb R^n$ , closed balls of radius $j$ suffice. Some initial sets can be empty.

Use

An exhaustion turns local requirements into a countable sequence of estimates. In smooth jet realization, the $j$ -th cutoff is chosen to control finitely many derivatives on $K_j$ .