In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space X is a nested sequence of compact subsets K_i of X (i.e. ), such that each K_i is contained in the interior of K_{i+1}, i.e., and. A space admitting an exhaustion by compact sets is called exhaustible by compact sets. As an example, for the space, the sequence of closed balls forms an exhaustion of the space by compact sets. There is a weaker condition that drops the requirement that K_i is in the interior of K_{i+1}, meaning the space is σ-compact (i.e., a countable union of compact subsets.)

Construction

If there is an exhaustion by compact sets, the space is necessarily locally compact (if Hausdorff). The converse is also often true. For example, for a locally compact Hausdorff space X that is a countable union of compact subsets, we can construct an exhaustion as follows. We write as a union of compact sets K_n. Then inductively choose open sets with compact closures, where. Then is a required exhaustion. For a locally compact Hausdorff space that is second-countable, a similar argument can be used to construct an exhaustion.

Application

For a Hausdorff space X, an exhaustion by compact sets can be used to show the space is paracompact. Indeed, suppose we have an increasing sequence of open subsets such that and each is compact and is contained in V_{n+1}. Let \mathcal{U} be an open cover of X. We also let. Then, for each n \ge 1, is an open cover of the compact set and thus admits a finite subcover. Then is a locally finite refinement of Remark: The proof in fact shows that each open cover admits a countable refinement consisting of open sets with compact closures and each of whose members intersects only finitely many others. The following type of converse also holds. A paracompact locally compact Hausdorff space with countably many connected components is a countable union of compact sets and thus admits an exhaustion by compact subsets.

Relation to other properties

The following are equivalent for a topological space X: (where weakly locally compact means locally compact in the weak sense that each point has a compact neighborhood). The hemicompact property is intermediate between exhaustible by compact sets and σ-compact. Every space exhaustible by compact sets is hemicompact and every hemicompact space is σ-compact, but the reverse implications do not hold. For example, the Arens-Fort space and the Appert space are hemicompact, but not exhaustible by compact sets (because not weakly locally compact), and the set \Q of rational numbers with the usual topology is σ-compact, but not hemicompact. Every regular Hausdorff space that is a countable union of compact sets is paracompact.