Exhaustion by compact subsets

In mathematics, especially 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 $X=\bigcup _{i=1}^{\mathrm{\infty }}{K}_i$. Sometimes the requirement that $K_i$ is in the interior of $K_{i+1}$ is dropped (and, in that case, the existence of an exhaustion by compact sets means the space is σ-compact space.)

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

Application: paracompactness

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". http://planetmath.org/?op=getobj&from=objects&id=7103.