Exhaustion by compact sets

In mathematics, especially general topology and analysis, an exhaustion by compact sets^[1] of a topological space $X$ is a nested sequence of compact subsets $K_{i}$ of $X$ ; i.e.

K_{1}\subset K_{2}\subset K_{3}\subset \cdots ,

such that $K_{i}$ is contained in the interior of $K_{i+1}$ , i.e. $K_{i}\subset {\text{int}}(K_{i+1})$ for each $i$ and $X=\bigcup _{i=1}^{\infty }K_{i}$ .

For example, consider $X=\mathbb {R} ^{n}$ and the sequence of closed balls $K_{i}=\{x:|x|\leq i\}.$ For a locally compact Hausdorff space $X$ that is a countable union of compact subsets, we can construct an exhaustion as follows. We write $X=\bigcup _{1}^{\infty }K_{n}$ as a union of compact sets $K_{n}$ . Then inductively choose open sets $V_{n}\supset {\overline {V_{n-1}}}\cup K_{n}$ with compact closures, where $V_{0}=\emptyset$ . Then ${\overline {V_{n}}}$ is a required exhaustion. For a locally compact Hausdorff space that is second-countable, a similar argument can be used to construct an exhaustion.

Occasionally some authors drop the requirement that $K_{i}$ is in the interior of $K_{i+1}$ , but then the property becomes the same as the space being σ-compact, namely a countable union of compact subsets. For example, $\mathbb {Q}$ is σ-compact but does not admit an exhaustion by compact sets since it is not locally compact.

Application

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

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.^[2]

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^[3] and thus admits an exhaustion by compact subsets.

Relation to other properties

A space admitting an exhaustion by compact sets is called exhaustible by compact sets.^{[citation needed]}

The following are equivalent for a topological space $X$ :^[4]

$X$ is exhaustible by compact sets.
$X$ is σ-compact and weakly locally compact.
$X$ 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^[5] 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),^[6] and the set $\mathbb {Q}$ of rational numbers with the usual topology is σ-compact, but not hemicompact.^[7]

Every regular space that is a countable union of compact sets is paracompact.^{[citation needed]}

Notes

^ Lee 2011, p. 110.
^ ^a ^b Warner, Ch. 1. Lemma 1.9.
^ Wall, Proposition A.2.8. (ii) NB: the proof in the reference looks problematic. It can be fixed by constructing an open cover whose member intersects only finitely many others. (Then we use the fact that a locally finite connected graph is countable.)
^ "A question about local compactness and $\sigma$-compactness". Mathematics Stack Exchange.
^ "Does locally compact and $\sigma$-compact non-Hausdorff space imply hemicompact?". Mathematics Stack Exchange.
^ "Can a hemicompact space fail to be weakly locally compact?". Mathematics Stack Exchange.
^ "A $\sigma$-compact but not hemicompact space?". Mathematics Stack Exchange.

References

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.
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.

External links

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

[FOOTNOTELee2011110-1] Lee 2011, p. 110.

[Warner-2] Warner, Ch. 1. Lemma 1.9.

[3] Wall, Proposition A.2.8. (ii) NB: the proof in the reference looks problematic. It can be fixed by constructing an open cover whose member intersects only finitely many others. (Then we use the fact that a locally finite connected graph is countable.)

[4] "A question about local compactness and $\sigma$-compactness". Mathematics Stack Exchange.

[5] "Does locally compact and $\sigma$-compact non-Hausdorff space imply hemicompact?". Mathematics Stack Exchange.

[6] "Can a hemicompact space fail to be weakly locally compact?". Mathematics Stack Exchange.

[7] "A $\sigma$-compact but not hemicompact space?". Mathematics Stack Exchange.

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