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# 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 $U$.

Related:

MethodOfExhaustion

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

53-00*no label found*

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