Editing Stone–Čech compactification (section)

===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]].)