Example: Inverse Stereographic Projection
A geometrically appealing example of one-point compactification is given by the inverse stereographic projection. Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the Euclidean plane. The inverse stereographic projection is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point . Under the stereographic projection latitudinal circles get mapped to planar circles . It follows that the deleted neighborhood basis of given by the punctured spherical caps corresponds to the complements of closed planar disks . More qualitatively, a neighborhood basis at is furnished by the sets 
as K ranges through the compact subsets of . This example already contains the key concepts of the general case.
Read more about this topic: Alexandroff Extension
Famous quotes containing the words inverse and/or projection:
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—Ralph Waldo Emerson (1803–1882)
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