Editing Stone–Čech compactification (section)

===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.