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 $K_n\subset \mathrm{int}{K}_{n+1}$ and ${\bigcup }_n{K}_n=U$. 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 $K_n$.