In mathematics, especially general topology and analysis, an exhaustion by compact sets[1] of a topological space
is a nested sequence of compact subsets
A space admitting an exhaustion by compact sets is called exhaustible by compact sets.
, the sequence of closed balls
forms an exhaustion of the space by compact sets.
There is a weaker condition that drops the requirement that
, meaning the space is σ-compact (i.e., a countable union of compact subsets.)
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
that is a countable union of compact subsets, we can construct an exhaustion as follows.
as a union of compact sets
Then inductively choose open sets
For a locally compact Hausdorff space that is second-countable, a similar argument can be used to construct an exhaustion.
, an exhaustion by compact sets can be used to show the space is paracompact.
[3] Indeed, suppose we have an increasing sequence
of open subsets such that
is an open cover of the compact set
and thus admits a finite subcover
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.
[3] 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[4] and thus admits an exhaustion by compact subsets.
The following are equivalent for a topological space
:[5] (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[6] 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),[7] and the set
of rational numbers with the usual topology is σ-compact, but not hemicompact.
[8] Every regular Hausdorff space that is a countable union of compact sets is paracompact.