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.