Exhaustion by compact sets

In mathematics, especially analysis, exhaustion by compact sets of an open set E in the Euclidean space Rⁿ (or a manifold with countable base) is an increasing sequence of compact sets $K_j$, where by increasing we mean $K_j$ is a subset of $K_{j+1}$, with the limit (union) of the sequence being E.

Sometimes one requires the sequence of compact sets to satisfy one more property— that $K_j$ is contained in the interior of $K_{j+1}$ for each $j$. This, however, is dispensed in Rⁿ or a manifold with countable base.

For example, consider a unit open disk and the concentric closed disk of each radius inside. That is let $E = \{ z; |z| < 1 \}$ and $K_j = \{ z; |z| \le (1 - 1/j) \}$. Then taking the limit (union) of the sequence $K_j$ gives E. The example can be easily generalized in other dimensions.

See also

• σ-compact space

References

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

External links

• Exhaustion by compact sets at PlanetMath.org.