Compactification

A compactification of a topological space  is a larger space  containing  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  has a compactification if and only if it is completely regular and a -space.

The extended real line  with the order topology is a two point compactification of . The projective plane can be viewed as a compactification of the plane.