Point at Infinity - Other Generalisations

Other Generalisations

This construction can be generalized to topological spaces. Different compactifications may exist for a given space, but arbitrary topological space admits Alexandroff extension, also called the one-point compactification when the original space is not itself compact. Projective line (over arbitrary field) is the Alexandroff extension of the corresponding field. Thus the circle is the one-point compactification of the real line, and the sphere is the one-point compactification of the plane. Projective spaces Pn for n > 1 are not one-point compactifications of corresponding affine spaces for the reason mentioned above, nor omega points are.