A compactification of a topological space X is a larger space Y containing X which is also compact. The smallest compactification is the one-point compactification. For example, the real line is not compact. It is contained in the circle, which is obtained by adding a point at infinity. Similarly, the plane is compactified by adding one point at infinity, giving the sphere. A topological space X has a compactification if and only if it is completely regular and a T_1-space. The extended real line R union {-∞, ∞} with the order topology is a two point compactification of R. The projective plane can be viewed as a compactification of the plane.

compact set | one-point compactification | stereographic projection | topological space