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.