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 (*X*_{1}, \(\cal T\!\)_{1})
and (*X*_{2}, \(\cal T\!\)_{2})
are compact topological spaces, then *X*_{1}
× *X*_{2} 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 {*x*_{n}}
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