Circle Group is Uncountably Infinite

Theorem
The circle group $\struct {K, \times}$ is an uncountably infinite group.

Proof
From Quotient Group of Reals by Integers is Circle Group, $\struct {K, \times}$ is isomorphic to the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$.

But $\dfrac {\struct {\R, +} } {\struct {\Z, +} }$ is the half-open interval $\hointr 0 1$.

A real interval is uncountable by (some result).

Hence the result.