Английская Википедия:Exhaustion by compact sets

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In mathematics, especially general topology and analysis, an exhaustion by compact setsШаблон:Sfn of a topological space <math>X</math> is a nested sequence of compact subsets <math>K_i</math> of <math>X</math> (i.e. <math>K_1\subseteq K_2\subseteq K_3\subseteq\cdots</math>), such that <math>K_i</math> is contained in the interior of <math>K_{i+1}</math>, i.e. <math>K_i\subseteq\text{int}(K_{i+1})</math> for each <math>i</math> and <math>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>X=\mathbb{R}^n</math> and the sequence of closed balls <math>K_i = \{ x : |x| \le i \}.</math>

Occasionally some authors drop the requirement that <math>K_i</math> is in the interior of <math>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>X</math>:

1. <math>X</math> is exhaustible by compact sets.
2. <math>X</math> is σ-compact and weakly locally compact.
3. <math>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 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), and the set <math>\Q</math> of rational numbers with the usual topology is σ-compact, but not hemicompact.

Every regular space exhaustible by compact sets is paracompact.

References

• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. Шаблон:ISBN.
• Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. Шаблон:ISBN.
• Шаблон:Cite book

External links

• Шаблон:Planetmath reference
• Шаблон:Cite web