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 , where by increasing we mean is a subset of , with the limit (union) of the sequence being E.
Sometimes one requires the sequence of compact sets to satisfy one more property—that is contained in the interior of for each . 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 and . Then taking the limit (union) of the sequence gives E. The example can be easily generalized in other dimensions.
See also
References
- Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN 0-8218-1221-1.
External links
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