Editing Compactification (mathematics) (section)

== Definition ==

An [[embedding]] of a topological space ''X'' as a [[dense set|dense]] subset of a compact space is called a '''compactification''' of ''X''. It is often useful to embed [[topological space]]s in [[compact space]]s, because of the special properties compact spaces have.

Embeddings into compact [[Hausdorff space]]s may be of particular interest. Since every compact Hausdorff space is a [[Tychonoff space]], and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space.  In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.

The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.

=== Alexandroff one-point compactification ===
{{main|One-point compactification}}
For any noncompact topological space ''X'' the ('''Alexandroff''') '''one-point compactification''' α''X'' of ''X'' is obtained by adding one extra point ∞ (often called a ''point at infinity'') and defining the [[open set]]s of the new space to be the open sets of ''X'' together with the sets of the form ''G''&nbsp;∪&nbsp;{{mset|∞}}, where ''G'' is an open subset of ''X'' such that <math>X \setminus G</math> is closed and compact. The one-point compactification of ''X'' is Hausdorff if and only if ''X'' is Hausdorff and [[locally compact]].<ref>{{citation|first=Pavel S.|last= Alexandroff|author-link=Pavel Alexandroff| journal=  [[Mathematische Annalen]] |volume= 92|issue=3–4 |year=1924|pages= 294–301|title= Über die Metrisation der im Kleinen kompakten topologischen Räume | url=https://eudml.org/doc/159072 | doi=10.1007/BF01448011 | jfm=50.0128.04 }}</ref>

=== Stone–Čech compactification ===
{{main|Stone–Čech compactification}}
Of particular interest are Hausdorff compactifications, i.e., compactifications in which the compact space is [[Hausdorff space|Hausdorff]]. A topological space has a Hausdorff compactification if and only if it is [[Tychonoff space|Tychonoff]]. In this case, there is a unique ([[up to]] [[homeomorphism]]) "most general" Hausdorff compactification, the [[Stone–Čech compactification]] of ''X'', denoted by ''βX''; formally, this exhibits the [[category (mathematics)|category]] of Compact Hausdorff spaces and continuous maps as a [[reflective subcategory]] of the category of Tychonoff spaces and continuous maps.

"Most general" or formally "reflective" means that the space ''βX'' is characterized by the [[universal property]] that any [[continuous function (topology)|continuous function]] from ''X'' to a compact Hausdorff space ''K'' can be extended to a continuous function from ''βX'' to ''K'' in a unique way. More explicitly, ''βX'' is a compact Hausdorff space containing ''X'' such that the [[subspace topology|induced topology]] on ''X'' by ''βX'' is the same as the given topology on ''X'', and for any continuous map {{nowrap|''f'' : ''X'' → ''K''}}, where ''K'' is a compact Hausdorff space, there is a unique continuous map {{nowrap|''g'' : ''βX'' → ''K''}} for which ''g'' restricted to ''X'' is identically&nbsp;''f''.

The Stone–Čech compactification can be constructed explicitly as follows: let ''C'' be the set of continuous functions from ''X'' to the closed interval {{nowrap|[0, 1]}}.  Then each point in ''X'' can be identified with an evaluation function on ''C''.  Thus ''X'' can be identified with a subset of {{nowrap|[0, 1]<sup>''C''</sup>}}, the space of ''all'' functions from ''C'' to {{nowrap|[0, 1]}}.  Since the latter is compact by [[Tychonoff's theorem]], the closure of ''X'' as a subset of that space will also be compact.  This is the Stone–Čech compactification.<ref>{{cite journal|first=Eduard|last= Čech|author-link=Eduard Čech| title=On bicompact spaces|journal=  [[Annals of Mathematics]] |volume= 38  |year=1937  |pages= 823–844|doi=10.2307/1968839|issue=4|jstor=1968839|hdl= 10338.dmlcz/100420|hdl-access=free}}</ref>
<ref>{{citation|first=Marshall H.|last= Stone|author-link=Marshall H. Stone|title=Applications of the theory of Boolean rings to general topology  |journal=[[Transactions of the American Mathematical Society]] |volume= 41  |year=1937|pages= 375–481
|issue=3|doi=10.2307/1989788 |jstor=1989788|doi-access=free}}</ref>

=== Spacetime compactification ===
[[Walter Benz]] and [[Isaak Yaglom]] have shown how [[stereographic projection]] onto a single-sheet [[hyperboloid]] can be used to provide a [[Motor variable#Compactification|compactification for split complex numbers]]. In fact, the hyperboloid is part of a [[quadric]] in real projective four-space. The method is similar to that used to provide a base manifold for [[group action (mathematics)|group action]] of the [[conformal group of spacetime]].<ref>15 parameter conformal group of spacetime described in {{Wikibooks-inline|Associative Composition Algebra/Homographies}}</ref>