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 {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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