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→Removing Points with Rigid Body Motions
We form the partition <math>\mathbb{S}^1 = T \cup (\mathbb{S}^1-T) \ </math> and then note <math>R_\theta(T) \cup (\mathbb{S}^1 - T) = \mathbb{S}^1 - A \ </math> and we have shown that <math>\mathbb{S}^1 \cong \mathbb{S}^1 - A \ </math>
We form the partition <math>\mathbb{S}^1 = T \cup (\mathbb{S}^1-T) \ </math> and then note <math>R_\theta(T) \cup (\mathbb{S}^1 - T) = \mathbb{S}^1 - A \ </math> and we have shown that <math>\mathbb{S}^1 \cong \mathbb{S}^1 - A \ </math>
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Removing countably many points from a sphere is similar, except now we are using the longitude of the point. A tricky technical matter arises:
Removing countably many points from a sphere is similar, except now we are using the longitudes of the points, with respect to some axis that doesn't pass through any of the points of <math>A \ </math>. A tricky technical matter arises: how do we know such an axis exists?
This is simple! Since <math>A \ </math> is countable, so is the union of <math>A \ </math> and all the antipodes of <math>A \ </math>, which means there are points not in this union. By design, neither are the antipodes of such points.
===The Free Group===
===The Free Group===