Compactifications of X may be ordered: we say that if there is a continuous map h of Y onto Z such that h.f = g.
The Stone-Čech compactification of X is constructed from the unit interval I. Let F(X) be the family of continuous maps from X to I and let the "cube" IF(X) be the Cartesian power with the product topology. The evaluation map e maps X to IF(X),regarded as the set of functions from F(X) to I, by
The evaluation map e is a continuous map from X to the cube and we let β(X) denote the closure of the image of e. The Stone-Čech compactification is then the pair (e,β(X)).
If we restrict attention to the partial order of Hausdorff compactifications, then the one-point compactification is the minimum and the Stone-Čech compactification is the maximum element for this order. The latter states that if X is a Tychonoff space then any continuous map from X to a compact space can be extended to a map from β(X) compatible with e. This extension property characterises the Stone-Čech compactification.
- J.L. Kelley (1955). General topology. van Nostrand, 149-156.