Editing Stone–Čech compactification (section)

===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.