Editing Stone–Čech compactification (section)

== Universal property and functoriality ==

The Stone–Čech compactification of the topological space ''X'' is a compact Hausdorff space  ''βX'' together with a continuous map ''i<sub>X</sub>'' : ''X'' → ''βX'' that has the following [[universal property]]: any [[continuous map]] ''f'' : ''X'' → ''K'', where ''K'' is a compact Hausdorff space, extends uniquely to a continuous map ''βf'' : ''βX'' → ''K'', i.e. ({{itco|''βf''}})''i<sub>X</sub>'' = ''f''.{{sfn|Munkres|2000|pp=239, Theorem 38.4}}
[[File:Stone cech diagram.svg|center|The universal property of the Stone-Cech compactification expressed in diagram form.|250px]]
As is usual for universal properties, this universal property characterizes ''βX'' [[up to]] [[homeomorphism]].

As is outlined in {{sectionlink||Constructions}}, below, one can prove (using the axiom of choice) that such a Stone–Čech compactification ''i<sub>X</sub>'' : ''X'' → ''βX'' exists for every topological space ''X''.  Furthermore, the image ''i<sub>X</sub>''(''X'') is dense in ''βX''.

Some authors add the assumption  that the starting space ''X'' be Tychonoff (or even [[locally compact]] Hausdorff), for the following reasons:
*The map from ''X''  to its image in  ''βX'' is a homeomorphism if and only if ''X'' is Tychonoff. 
*The map from ''X'' to its image  in ''βX'' is a homeomorphism to an open subspace  if and only if ''X'' is locally compact Hausdorff.
The Stone–Čech construction can be performed for more general spaces ''X'', but in that case the map ''X'' → ''βX'' need not be a homeomorphism to the image of ''X''  (and sometimes is not even injective).

As is usual for universal constructions like this, the extension property makes ''β'' a [[functor]] from '''Top''' (the [[category of topological spaces]]) to '''CHaus''' (the category of compact Hausdorff spaces). Further, if we let ''U'' be the [[inclusion functor]] from '''CHaus''' into '''Top''', maps from ''βX'' to ''K'' (for ''K'' in '''CHaus''') correspond [[Bijection|bijectively]] to maps from ''X'' to ''UK'' (by considering their [[Restriction (mathematics)|restriction]] to ''X'' and using the universal property of ''βX''). i.e. 
:Hom(''βX'', ''K'') ≅ Hom(''X'', ''UK''), 
which means that ''β'' is [[adjoint functor|left adjoint]] to ''U''. This implies that '''CHaus''' is a [[reflective subcategory]] of '''Top''' with reflector ''β''.