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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20 feb 2016