# 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 ${\displaystyle K_{j}}$, where by increasing we mean ${\displaystyle K_{j}}$ is a subset of ${\displaystyle K_{j+1}}$, with the limit (union) of the sequence being E.

Sometimes one requires the sequence of compact sets to satisfy one more property— that ${\displaystyle K_{j}}$ is contained in the interior of ${\displaystyle K_{j+1}}$ for each ${\displaystyle j}$. 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 ${\displaystyle E=\{z;|z|<1\}}$ and ${\displaystyle K_{j}=\{z;|z|\leq (1-1/j)\}}$. Then taking the limit (union) of the sequence ${\displaystyle K_{j}}$ gives E. The example can be easily generalized in other dimensions.