Editing Stone–Čech compactification (section)

==Constructions==

===Construction using products===
One attempt to construct the Stone–Čech compactification of ''X'' is to take the closure of the image of ''X'' in

:<math>\prod\nolimits_{f:X\to K} K</math>

where the product is over all maps from ''X'' to compact Hausdorff spaces ''K'' (or, equivalently, the image of ''X'' by the right [[Kan extension]] of the identity functor of the category ''CHaus'' of compact Hausdorff spaces along the inclusion functor of ''CHaus'' into the category ''Top'' of general topological spaces).<ref group=Note>Refer to Example 4.6.12 for an explicit left adjoint construction, or to Proposition 6.5.2 for how left adjoints can be seen as right Kan extensions in {{cite book | author=Riehl | title=Category Theory in Context |year=2014|page=149, 210}}</ref> By [[Tychonoff's theorem]] this product of compact spaces is compact, and the closure of ''X'' in this space is therefore also compact. This works intuitively but fails for the technical reason that  the collection of all such maps is a [[proper class]] rather than a set. There are several ways to modify this idea to make it work; for example, one can restrict the compact Hausdorff spaces ''K'' to have underlying set ''P''(''P''(''X'')) (the [[power set]] of the power set of ''X''), which is sufficiently large that it has cardinality at least equal to that of every compact Hausdorff space to which ''X'' can be mapped with dense image.

===Construction using the unit interval===
One way of constructing ''βX'' is to let ''C'' be the set of all [[continuous function]]s from ''X'' into [0, 1] and consider the map <math> e: X \to [0,1]^{C} </math> where 

:<math> e(x): f \mapsto f(x) </math>

This may be seen to be a continuous map onto its image, if [0, 1]<sup>''C''</sup> is given the [[product topology]]. By [[Tychonoff's theorem]] we have that [0, 1]<sup>''C''</sup> is compact since [0, 1] is. Consequently, the closure of ''X'' in [0, 1]<sup>''C''</sup> is a compactification of ''X''.

In fact, this closure is the Stone–Čech compactification. To verify this, we just need to verify that the closure satisfies the appropriate universal property. We do this first for ''K'' = [0, 1], where the desired extension of ''f'' : ''X'' → [0, 1] is just the projection onto the ''f'' coordinate in [0, 1]<sup>''C''</sup>. In order to then get this for general compact Hausdorff ''K'' we use the above to note that ''K'' can be embedded in some cube, extend each of the coordinate functions and then take the product of these extensions.

The special property of the [[unit interval]] needed for this construction to work is that it is a ''cogenerator'' of the category of compact Hausdorff spaces: this means that if ''A'' and ''B'' are compact Hausdorff spaces, and ''f'' and ''g'' are distinct maps from ''A'' to ''B'', then there is a map ''h'' : ''B'' → [0, 1] such that ''hf'' and ''hg'' are distinct. Any other cogenerator (or cogenerating set) can be used in this construction.

===Construction using ultrafilters===
{{See also|Stone topology|Filters in topology#Stone topology}}

Alternatively, if {{mvar|X}} is [[Discrete space|discrete]], then it is possible to construct <math>\beta X</math> as the set of all [[Ultrafilter (set theory)|ultrafilter]]s on {{mvar|X}}, with the elements of {{mvar|X}} corresponding to the [[Ultrafilter|principal ultrafilter]]s. The topology on the set of ultrafilters, known as the {{em|[[Stone topology|{{visible anchor|Stone topology|Ultrafilter construction}}]]}}, is generated by sets of the form <math>\{ F : U \in F \}</math> for {{mvar|U}} a subset of {{mvar|X}}.

Again we verify the universal property: For <math>f : X \to K</math> with {{mvar|K}} compact Hausdorff and {{mvar|F}} an ultrafilter on {{mvar|X}} we have an [[Filter (set theory)#Ultrafilters|ultrafilter base]] <math>f(F)</math> on {{mvar|K}}, the [[Pushforward (differential)|pushforward]] of {{mvar|F}}. This has a unique [[Limit (mathematics)|limit]] because {{mvar|K}} is compact Hausdorff, say {{mvar|x}}, and we define <math>\beta f(F) = x.</math> This may be verified to be a continuous extension of {{mvar|f}}.

Equivalently, one can take the [[Stone space]] of the [[complete Boolean algebra]] of all subsets of {{mvar|X}} as the Stone–Čech compactification. This is really the same construction, as the Stone space of this Boolean algebra is the set of ultrafilters (or equivalently prime [[Ideal (order theory)|ideal]]s, or homomorphisms to the 2-element Boolean algebra) of the Boolean algebra, which is the same as the set of ultrafilters on {{mvar|X}}.

The construction can be generalized to arbitrary Tychonoff spaces by using [[Maximal filter|maximal filters]] of [[zero set]]s instead of ultrafilters.<ref>
W.W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', Springer, 1974.</ref> (Filters of closed sets suffice if the space is [[normal space|normal]].)

===Construction using C*-algebras===
The Stone–Čech compactification is naturally homeomorphic to the [[Spectrum of a C*-algebra|spectrum]] of C<sub>b</sub>(''X'').<ref>This is Stone's original construction.</ref> Here C<sub>b</sub>(''X'') denotes the [[C*-algebra]] of all continuous bounded [[Complex-valued function|complex-valued functions]] on ''X'' with [[Sup norm|sup-norm]]. Notice that C<sub>b</sub>(''X'') is canonically isomorphic to the [[multiplier algebra]] of C<sub>0</sub>(''X'').