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 $$(f,Y) \ge (h,Z)$$ 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" 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


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

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)).