In mathematics, especially general topology and analysis, an exhaustion by compact sets^[1] of a topological space ${\displaystyle X}$ is a nested sequence of compact subsets ${\displaystyle K_{i}}$ of ${\displaystyle X}$ (i.e. ${\displaystyle K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots }$), such that each ${\displaystyle K_{i}}$ is contained in the interior of ${\displaystyle K_{i+1}}$, i.e. ${\displaystyle K_{i}\subset {\text{int}}(K_{i+1})}$, and ${\displaystyle X=\bigcup _{i=1}^{\infty }K_{i}}$.

A space admitting an exhaustion by compact sets is called exhaustible by compact sets.^[2]

As an example, for the space ${\displaystyle X=\mathbb {R} ^{n}}$, the sequence of closed balls ${\displaystyle K_{i}=\{x:|x|\leq i\}}$ forms an exhaustion of the space by compact sets.

There is a weaker condition that drops the requirement that ${\displaystyle K_{i}}$ is in the interior of ${\displaystyle K_{i+1}}$, 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 ${\displaystyle X}$ that is a countable union of compact subsets, we can construct an exhaustion as follows. We write ${\displaystyle X=\bigcup _{1}^{\infty }K_{n}}$ as a union of compact sets ${\displaystyle K_{n}}$. Then inductively choose open sets ${\displaystyle V_{n}\supset {\overline {V_{n-1}}}\cup K_{n}}$ with compact closures, where ${\displaystyle V_{0}=\emptyset }$. Then ${\displaystyle {\overline {V_{n}}}}$ is a required exhaustion.

For a locally compact Hausdorff space that is second-countable, a similar argument can be used to construct an exhaustion.

For a Hausdorff space ${\displaystyle X}$, an exhaustion by compact sets can be used to show the space is paracompact.^[3] Indeed, suppose we have an increasing sequence ${\displaystyle V_{1}\subset V_{2}\subset \cdots }$ of open subsets such that ${\displaystyle X=\bigcup V_{n}}$ and each ${\displaystyle {\overline {V_{n}}}}$ is compact and is contained in ${\displaystyle V_{n+1}}$. Let ${\displaystyle {\mathcal {U}}}$ be an open cover of ${\displaystyle X}$. We also let ${\displaystyle V_{n}=\emptyset ,\,n\leq 0}$. Then, for each ${\displaystyle n\geq 1}$, ${\displaystyle \{(V_{n+1}-{\overline {V_{n-2}}})\cap U\mid U\in {\mathcal {U}}\}}$ is an open cover of the compact set ${\displaystyle {\overline {V_{n}}}-V_{n-1}}$ and thus admits a finite subcover ${\displaystyle {\mathcal {V}}_{n}}$. Then ${\displaystyle {\mathcal {V}}:=\bigcup _{n=1}^{\infty }{\mathcal {V}}_{n}}$ is a locally finite refinement of ${\displaystyle {\mathcal {U}}.}$

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 ${\displaystyle X}$:^[5]

1. ${\displaystyle X}$ is exhaustible by compact sets.
2. ${\displaystyle X}$ is σ-compact and weakly locally compact.
3. ${\displaystyle X}$ is Lindelöf and weakly locally compact.

(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 ${\displaystyle \mathbb {Q} }$ 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.^{[citation needed]}

• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN 0-8218-1221-1.
• Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. ISBN 978-3540003731.
• Harder, Günter (2011). Lectures on algebraic geometry. 1: Sheaves, cohomology of sheaves, and applications to Riemann surfaces (2nd ed.). ISBN 3834818445.
• Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7939-1.
• Warner, Frak W. (1983). Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics. Springer-Verlag.
• Wall, C. T. C. Differential Topology. Cambridge University Press. ISBN 9781107153523.

• "Exhaustion by compact sets". PlanetMath.
• "Existence of exhaustion by compact sets". Mathematics Stack Exchange.