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 Kj, where by increasing we mean Kj is a subset of Kj + 1, with the limit (union) of the sequence being E. Sometimes one requires the sequence of compact sets to satisfy one more property— that Kj is contained in the interior of Kj + 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 Kj gives E. The example can be easily generalized in other dimensions.

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