exhaustion by compact sets

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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 $U$ .

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Theorem

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