Finally, connect every point on S2 with a ray to the origin; the paradoxical decomposition of S2 then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's centre (this center point needs a bit more care, see below).
N.B. This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen to lie on the axis of some rotation in H. However, there are only countably many such points, and like the point at the centre of the ball, it is possible to patch the proof to account for them all (see below).
Famous quotes containing the word step:
“It is a good lessonthough it may often be a hard onefor a man who has dreamed of literary fame, and of making for himself a rank among the worlds dignitaries by such means, to step aside out of the narrow circle in which his claims are recognized, and to find how utterly devoid of all significance, beyond that circle, is all that he achieves, and all he aims at.”
—Nathaniel Hawthorne (18041864)