# exhaustion by compact sets

Let $U$ be an open set in $\mathbbmss{R}^{n}$ (or a manifold with countable base).  Then there exists a sequence of compact sets $K_{1},K_{2},\ldots$ such that

 $\displaystyle K_{i}$ $\displaystyle\subseteq$ $\displaystyle\operatorname{int}K_{i+1},\quad i=1,2,\ldots,$ $\displaystyle U$ $\displaystyle=$ $\displaystyle\cup_{i=1}^{\infty}K_{i},$

where “$\operatorname{int}$” denotes the topological interior.   Such a sequence is called an exhaustion by compact sets for $U$.

Title exhaustion by compact sets ExhaustionByCompactSets 2013-03-22 15:18:13 2013-03-22 15:18:13 matte (1858) matte (1858) 6 matte (1858) Theorem msc 53-00 MethodOfExhaustion