Exhaustion by compact sets

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 on PlanetMath

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