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→The Banach-Tarski Construction
===The Banach-Tarski Construction===
===The Banach-Tarski Construction===
We begin by removing the center of the unit ball by choosing some circle in the ball which contains the center, and using the method by which we removed a point from the circle earlier. Now that we have done this, we can work on the surface of the ball, <math>\mathbb{S}^2 \ </math>, by understanding that everything we do on the surface can extend downward to the interior of the ball in a ray towards the center.
Now we define the '''orbit''' of a point ''x'' of the sphere under the operations of the free group to be <math>\mathbb{F}(x) \ </math>. For almost all the points on the surface, this will yield a copy of the free group, except a couple of bad points: for example, the "north pole" is fixed under the ''b'' operation. So define <math>B=\left\{{x\in\mathbb{S}^2: \exists r\in\mathbb{F}, (r\neq e) \and (r(x)=x) }\right\} \ </math>, that is, the set of all the points on the sphere which are held still by some rotation in <math>F_2 \ </math>.
Since B is countable, and we can remove any countable set of points from the sphere, remove this set: <math>\mathbb{S}^2 - B \ </math>.
==Interpretation==
==Interpretation==