= Compact exhaustion
A compact exhaustion of an open subset $U$ of a finite-dimensional <Euclidean space> is an increasing sequence of compact subsets $K_n\subset U$ with $K_n\subset\operatorname{int}K_{n+1}$ and $\bigcup_nK_n=U$. For example, choose smoothed bounds or suitable nested levels of distance from the complement and distance from the origin. Exit times from these sets express the lifetime of a <maximal local solution of a stochastic differential equation>; every finite-time continuous path lying in $U$ has compact image and is contained in some $K_n$.
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