In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space is a nested sequence of compact subsets of (i.e. ), such that is contained in the interior of , i.e. for each and . A space admitting an exhaustion by compact sets is called exhaustible by compact sets.
For example, consider and the sequence of closed balls
Occasionally some authors drop the requirement that is in the interior of , 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 :
is exhaustible by compact sets. is σ-compact and weakly locally compact. 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 of rational numbers with the usual topology is σ-compact, but not hemicompact.
Every regular space exhaustible by compact sets is paracompact.
Notes
References
Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. . Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. .
External links
Compactness (mathematics) Mathematical analysis General topology