Automoderated users, edit
3,552
edits
Changes
Jump to navigation
Jump to search
Line 77:
Line 77: + +
→The Banach-Tarski Construction
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>.
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>.
Now we define an orbit equivalence relation: for two points <math>x,y \in \mathbb{S}^2-B \ </math>, define <math>x \sim y \ </math> whenever <math>x \ </math> and <math>y \ </math> are in the same orbit, ie, <math>\exists r \in \mathbb{F}:r(x)=y \ </math>. We state without proof that this ''is'' an equivalence relation and thus partitions <math>\mathbb{S}^2-B \ </math> into equivalence classes, which will be disjoint.
==Interpretation==
==Interpretation==