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 $K_{j}=\{z;|z|\leq (1-1/j)\}$ . Then taking the limit (union) of the sequence $K_{j}$ gives E. The example can be easily generalized in other dimensions.

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". PlanetMath.

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Exhaustion by compact sets

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References

External links