## Compactifying the xy Plane

Letbe theplane as a subset ofwith the Euclidean topology. Letbe the sphere of radius 1 with centre atThenis a compactification of

Letbe the line passing through the pointon the sphereTake any pointin the plane and draw a line throughandThis line will intersect the sphere at a point We can define a functionfrom the plane to the sphere in this way.

is one to one and ontoand bothandare continuous. Henceis a homeomorphism from the plane to the subsetof

is not compact whileis compact, and the setis dense inso thatis a compactification of