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 | ≤ ( 1 − 1 / j ) } . Then taking the limit (union) of the sequence K j gives E. The example can be easily generalized in other dimensions.