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
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