A compactification of a topological space X is a larger space Y containing X 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 X has a compactification if and only if it is completely regular and a T_1-space.

The extended real line R union {-infty,infty} with the order topology is a two point compactification of R. The projective plane can be viewed as a compactification of the plane.

See also

Compact Set, One-Point Compactification, Stereographic Projection, Topological Space

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Allan Cortzen

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Cite this as:

Cortzen, Allan; Rowland, Todd; and Weisstein, Eric W. "Compactification." From MathWorld--A Wolfram Web Resource.

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