Exhaustion by compact sets
In mathematics, especially general topology and analysis, an exhaustion by compact sets[1] 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]. A space admitting an exhaustion by compact sets is called exhaustible by compact sets. 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]
Occasionally some authors drop the requirement that [math]\displaystyle{ K_i }[/math] is in the interior of [math]\displaystyle{ K_{i+1} }[/math], but then the property becomes the same as the space being σ-compact, namely a countable union of compact subsets.
Properties
The following are equivalent for a topological space [math]\displaystyle{ X }[/math]:[2]
- [math]\displaystyle{ X }[/math] is exhaustible by compact sets.
- [math]\displaystyle{ X }[/math] is σ-compact and weakly locally compact.
- [math]\displaystyle{ X }[/math] is Lindelöf and weakly locally compact.
(where weakly locally compact means locally compact in the weak sense that each point has a compact neighborhood).
The hemicompact property is intermediate between exhaustible by compact sets and σ-compact. Every space exhaustible by compact sets is hemicompact[3] and every hemicompact space is σ-compact, but the reverse implications do not hold. For example, the Arens-Fort space and the Appert space are hemicompact, but not exhaustible by compact sets (because not weakly locally compact),[4] and the set [math]\displaystyle{ \Q }[/math] of rational numbers with the usual topology is σ-compact, but not hemicompact.[5]
Every regular space exhaustible by compact sets is paracompact.[6]
Notes
- ↑ Lee 2011, p. 110.
- ↑ "A question about local compactness and $\sigma$-compactness". https://math.stackexchange.com/q/2078142.
- ↑ "Does locally compact and $\sigma$-compact non-Hausdorff space imply hemicompact?". https://math.stackexchange.com/a/4569500/52912.
- ↑ "Can a hemicompact space fail to be weakly locally compact?". https://math.stackexchange.com/questions/4566624.
- ↑ "A $\sigma$-compact but not hemicompact space?". https://math.stackexchange.com/questions/4209303.
- ↑ "locally compact and sigma-compact spaces are paracompact in nLab". https://ncatlab.org/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact.
References
- Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN:0-8218-1221-1.
- Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. ISBN:978-3540003731.
- Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7939-1.
External links
- "Exhaustion by compact sets". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.
- "Existence of exhaustion by compact sets". https://math.stackexchange.com/questions/1360900.
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