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.

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Allan Cortzen

CITE THIS AS:

Cortzen, Allan; Rowland, Todd; and Weisstein, Eric W. "Compactification." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Compactification.html