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. $K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots$ ), such that $K_{i}$ is contained in the interior of $K_{i+1}$ , i.e. $K_{i}\subseteq {\text{int}}(K_{i+1})$ for each $i$ and $X=\bigcup _{i=1}^{\infty }K_{i}$ . A space admitting an exhaustion by compact sets is called exhaustible by compact sets.

For example, consider $X=\mathbb {R} ^{n}$ and the sequence of closed balls $K_{i}=\{x:|x|\leq i\}.$

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.

Properties

The following are equivalent for a topological space $X$ :^[2]

$X$ 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^[3] 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),^[4] and the set $\mathbb {Q}$ of rational numbers with the usual topology is σ-compact, but not hemicompact.^[5]

Every regular space exhaustible by compact sets is paracompact.^[6]

Notes

^ Lee 2011, p. 110.
^ "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.
^ "locally compact and sigma-compact spaces are paracompact in nLab". ncatlab.org.

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.