Exhaustion by compact subsets

In mathematics, especially general topology and analysis, an exhaustion by compact subsets 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

A topological space is exhaustible by compact sets if and only if it is σ-compact and locally compact (in the sense that each point has a compact neighborhood).

An exhaustion by compact subsets can be used to show the space is paracompact.

Further reading

• Chill2Macht (https://math.stackexchange.com/users/327486/chill2macht), Existence of exhaustion by compact sets, URL (version: 2022-02-14): https://math.stackexchange.com/q/4381395

References

• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN 0-8218-1221-1.

• John Lee, Introduction to Topological Manifolds, Springer Verlag, 2nd ed. 2011. ISBN 978-1441979391.

• Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. ISBN 978-3540003731.

External links

• "Exhaustion by compact sets". PlanetMath.