Exhaustion by compact subsets
In mathematics, especially analysis, an exhaustion by compact subsets of a topological space [math]\displaystyle{ X }[/math] is a nested sequence of compact subsets [math]\displaystyle{ K_i }[/math] of [math]\displaystyle{ X }[/math] (i.e. [math]\displaystyle{ K_1\subseteq K_2\subseteq K_3\subseteq\cdots }[/math]), such that [math]\displaystyle{ K_i }[/math] is contained in the interior of [math]\displaystyle{ K_{i+1} }[/math] , i.e. [math]\displaystyle{ K_i\subseteq\text{int}(K_{i+1}) }[/math] for each [math]\displaystyle{ i }[/math] and [math]\displaystyle{ X=\bigcup_{i=1}^\infty K_i }[/math]. Sometimes the requirement that [math]\displaystyle{ K_i }[/math] is in the interior of [math]\displaystyle{ K_{i+1} }[/math] is dropped (and, in that case, the existence of an exhaustion by compact sets means the space is σ-compact space.)
For example, consider [math]\displaystyle{ X= {\mathbb R}^n }[/math] and the sequence of closed balls [math]\displaystyle{ K_i = \{ x : |x| \le i \} }[/math].
Application: paracompactness
An exhaustion by compact subsets can be used to show the space is paracompact.[citation needed]
Further reading
- Chill2Macht (https://math.stackexchange.com/users/327486/chill2macht), Existence of exhaustion by compact sets, URL (version: 2022-02-14): https://math.stackexchange.com/q/4381395
References
- Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN:0-8218-1221-1.
- John Lee, Introduction to Topological Manifolds, Springer Verlag, 2nd ed. 2011. ISBN:978-1441979391.
- Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. ISBN:978-3540003731.
External links
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