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 $(f,Y)\geq (h,Z)$ 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 $X^{*}=X\sqcup \{\omega \}$ where the neighbourhoods of ω are of the form $N_{K}(\omega )=\{\omega \}\cup (X\setminus K)$ 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

$e:x\mapsto (f\mapsto f(x)).\,$

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.