Theorem 1

The (multiplicative, topological group $U$ of complex numbers of absolute value 1 is isomorphic to the (additive, topological group $T$ of real numbers mod 1.

$U = S_1$ is compact and connected, and has a neighbourhood of the identity element -1 homeomorphic to an open interval of $\mathbf{R}$ (Chapter VI, § 2, no. 4, Proposition 5); the theorem is therefore a consequence of the topological characterization of $T$ given in Chapter V, § 2, Theorem 2.