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In [[mathematics]], especially [[mathematical analysis|analysis]], '''exhaustion by compact sets''' of an [[open set]] ''E'' in the [[Euclidean space]] '''R'''<sup>''n''</sup> (or a [[manifold]] with [[Second countable|countable base]]) is an increasing [[sequence]] of [[compact set]]s <math>K_j</math>, where by increasing we mean <math>K_j</math> is a [[subset]] of <math>K_{j+1}</math>, with the limit ([[set union|union]]) of the sequence being ''E''.

Sometimes one requires the sequence of compact sets to satisfy one more property— that <math>K_j</math> is contained in the [[Interior (topology)|interior]] of <math>K_{j+1}</math> for each <math>j</math>. This, however, is dispensed in '''R'''<sup>''n''</sup> 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 <math>E = \{ z; |z| < 1 \}</math> and <math>K_j = \{ z; |z| \le (1 - 1/j) \}</math>. Then taking the limit (union) of the sequence <math>K_j</math> 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==

* {{planetmath reference|title=Exhaustion by compact sets|id=7103}}

[[Category:Compactness (mathematics)]]
[[Category:Mathematical analysis]]

{{Mathanalysis-stub}}

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