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Date: 24-7-2021
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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.
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