Compactness |
In General
$ Def: A topological space
(X, \(\cal T\)) is compact if every open cover of X has
a finite subcover.
* Other characterization:
In terms of nets (see the Bolzano-Weierstrass theorem below); In terms of
filters, dual to covers (the topological space is compact if every filter
base has a cluster/adherent point; every ultrafilter is convergent).
* Results: A closed subspace of
a compact space is compact; A compact subspace of a Hausdorff space is closed.
> Online resources:
see Wikipedia page.
And Operations on Topologies
* Tychonoff theorem:
If (X1, \(\cal T\!\)1)
and (X2, \(\cal T\!\)2)
are compact topological spaces, then X1
× X2 is compact with respect to the product topology;
Remains true when generalized to products of arbitrary cardinality, but its proof for infinitely many
spaces requires the use of the axiom of choice.
Local Compactness (In the Strong Sense)
$ Def: A topological
space X is locally compact if for all x ∈ X and
all open neighborhoods U of x, there is another neighborhood
V whose closure is compact and contained in U.
Precompactness
$ For a topological space:
A subset Y is precompact in (X, \(\cal T\)) if every
sequence in Y has a subsequence that converges in X.
$ For a metric space: The
metric space (X, d) is precompact if for all ε
> 0 there is a finite cover of X by sets of diameter <
ε (or there is a finite subset F with
d(x, F) < ε for all
x ∈ X).
@ References: Dieudonné 69, v1, #16.
Other Types, Concepts, and Results > s.a. Bicompact Space;
paracompact space; types of topologies.
* Other types, generalizations:
Countable compactness, paracompactness, metacompactness, Lindelöf spaces.
* Bolzano-Weierstraß theorem: A Hausdorff space is compact iff
every net has a convergent subnet; More precisely, if (X, \(\cal T\))
is a topological space and A a subset of X, then
- If A is compact,
then each sequence {xn}
of points in A has an accumulation point in A;
- If A is second countable
and each sequence of points in A has
an accumulation point in A, then A is compact.
* Heine-Borel theorem:
A subset S ⊂ \(\mathbb R\) is
compact iff it is closed and bounded, i.e., of the form [a, b]
or a finite union thereof; In \(\mathbb R\)n,
the compact subsets are generated by products of subsets of \(\mathbb R\) of the type above.
@ References: Sanders CQG(13)-a1211 [spacelike and timelike compactness of a spacetime subset].
Compactification of a Space > s.a. asymptotic flatness;
Bohr Compactification; Spacetime Compactification.
* Rem: Different compactification
methods are available (e.g., one-point compactification, Stone-Cech compactification,
Wallman compactification, Fan-Gottesman compactification).
* End: A point added to compactify
a non-compact manifold, one for each essentially distinct way of going to infinity,
Introduced by H Freudenthal in 1930; For example, 2 ends for \(\mathbb R\) make
it homeomorphic to I = [0,1]; End theorem: It establishes criteria for being
able to add a boundary to a non-compact manifold to make it compact.
@ References: Torre CQG(04)gq [and group cohomology];
Elmali IJGMP(10)
[relations among compactification methods for locally compact Hausdorff spaces].
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send feedback and suggestions to bombelli at olemiss.edu – modified 20 feb 2016