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.
