A Hausdorff compactification of a topological space 
is a compact T2-space 
in which 
is embedded as a dense
subspace. A noncompact locally compact T2-space
has a one-point compactification, which
is its smallest Hausdorff compactification in the usual domination order. For example,
the real line is densely embedded in the circle by adding
a point at infinity. Similarly, adding one point at infinity compactifies the plane
to the sphere.
A topological space 
has a Hausdorff compactification iff
it is completely regular and a T1-space.
The Hausdorff one-point compactification
requires the space to be locally
compact; it does not exist for every such 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.
