Exhaustion by compact sets

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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| \le (1 - 1/j) \}. Then taking the limit (union) of the sequence K_j gives E. The example can be easily generalized in other dimensions.

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