Editing Alexandroff extension (section)

== 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 <math>S^{-1}: \mathbb{R}^2 \hookrightarrow S^2</math> is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point <math>\infty = (0,0,1)</math>.  Under the stereographic projection latitudinal circles <math>z = c</math> get mapped to planar circles <math display=inline>r = \sqrt{(1+c)/(1-c)}</math>.  It follows that the deleted neighborhood basis of <math>(0,0,1)</math> given by the punctured spherical caps <math>c \leq z < 1</math> corresponds to the complements of closed planar disks <math display=inline>r \geq \sqrt{(1+c)/(1-c)}</math>.  More qualitatively, a neighborhood basis at <math>\infty</math> is furnished by the sets <math>S^{-1}(\mathbb{R}^2 
\setminus K) \cup \{ \infty \}</math> as ''K'' ranges through the compact subsets of <math>\mathbb{R}^2</math>.  This example already contains the key concepts of the general case.