Changes

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.

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.  Note there will be uncountably many orbits, since each only has countably many points and the union of all orbits covers the uncountable set <math>\mathbb{S}^2 - B \ </math>.

 

 

Make sure you understand the notation: <math>\mathbb{F} \ </math> is a set of rotations, <math>W \ </math> is a set of points, and <math>\mathbb{F}(W) \ </math> is the set of points resulting from applying every rotation in <math>\mathbb{F} \ </math> to every point in <math>W \ </math>.

==Interpretation==

==Interpretation==