# exhaustion by compact sets

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

${K}_{i}$ | $\subseteq $ | $\mathrm{int}{K}_{i+1},i=1,2,\mathrm{\dots},$ | ||

$U$ | $=$ | ${\cup}_{i=1}^{\mathrm{\infty}}{K}_{i},$ |

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

Title | exhaustion by compact sets |
---|---|

Canonical name | ExhaustionByCompactSets |

Date of creation | 2013-03-22 15:18:13 |

Last modified on | 2013-03-22 15:18:13 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 6 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 53-00 |

Related topic | MethodOfExhaustion |