# Compactification

In general topology, a **compactification** of a topological space is a compact space in which the original space can be embedded, allowing the space to be studied using the properties of compactness.

Formally, a compactification of a topological space *X* is a pair (*f*,*Y*) where *Y* is a compact topological space and *f*:*X* → *Y* is a homeomorphism from *X* to a dense subset of *Y*.

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 **one-point compactification** of *X* is the disjoint union where the neighbourhoods of ω are of the form for *K* a closed compact subset of *X*.

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" *I*^{F(X)} be the Cartesian power with the product topology. The evaluation map *e* maps *X* to *I*^{F(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.

## References

- J.L. Kelley (1955).
*General topology*. van Nostrand, 149-156.