Exhaustion by compact sets explained
is a nested
sequence of compact subsets
of
(i.e.
K1\subseteqK2\subseteqK3\subseteq …
), such that each
is contained in the
interior of
, i.e.
, and
.
A space admitting an exhaustion by compact sets is called exhaustible by compact sets.
As an example, for the space
, the sequence of
closed balls
forms an exhaustion of the space by compact sets.
There is a weaker condition that drops the requirement that
is in the interior of
, meaning the space is
σ-compact (i.e., a
countable union of compact subsets.)
Construction
If there is an exhaustion by compact sets, the space is necessarily locally compact (if Hausdorff). The converse is also often true. For example, for a locally compact Hausdorff space
that is a countable union of compact subsets, we can construct an exhaustion as follows. We write
as a union of compact sets
. Then inductively choose open sets
} \cup K_n with compact closures, where
. Then
is a required exhaustion.
For a locally compact Hausdorff space that is second-countable, a similar argument can be used to construct an exhaustion.
Application
For a Hausdorff space
, an exhaustion by compact sets can be used to show the space is paracompact. Indeed, suppose we have an increasing sequence
of open subsets such that
and each
is compact and is contained in
. Let
be an open cover of
. We also let
. Then, for each
,
}) \cap U \mid U \in \mathcal \} is an open cover of the compact set
and thus admits a finite subcover
. Then
is a locally finite refinement of
Remark: The proof in fact shows that each open cover admits a countable refinement consisting of open sets with compact closures and each of whose members intersects only finitely many others.
The following type of converse also holds. A paracompact locally compact Hausdorff space with countably many connected components is a countable union of compact sets[1] and thus admits an exhaustion by compact subsets.
Relation to other properties
The following are equivalent for a topological space
:
[2]
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[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
of
rational numbers with the usual
topology is σ-compact, but not hemicompact.
[5] Every regular Hausdorff space that is a countable union of compact sets is paracompact.
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. .
- Book: Harder . Günter . Lectures on algebraic geometry. 1: Sheaves, cohomology of sheaves, and applications to Riemann surfaces . 2011 . 3834818445 . 2nd.
- Book: Lee . John M. . John M. Lee . Introduction to topological manifolds . 2011 . Springer . New York . 978-1-4419-7939-1 . 2nd.
- Book: Warner, Frak W.. Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics. 1983. Springer-Verlag.
- Book: Wall, C. T. C.. Differential Topology. Cambridge University Press. 9781107153523.
External links
Notes and References
- NB: the proof in the reference looks problematic. It can be fixed by constructing an open cover whose member intersects only finitely many others. (Then we use the fact that a locally finite connected graph is countable.)
- Web site: A question about local compactness and $\sigma$-compactness . Mathematics Stack Exchange.
- Web site: Does locally compact and $\sigma$-compact non-Hausdorff space imply hemicompact? . Mathematics Stack Exchange.
- Web site: Can a hemicompact space fail to be weakly locally compact? . Mathematics Stack Exchange.
- Web site: A $\sigma$-compact but not hemicompact space? . Mathematics Stack Exchange.
