Solution
= Solution
The topological <group> $S^1=\{z\in\mathbb C:|z|=1\}$ has complex multiplication and the subspace topology. It is compact and abelian, so every finite-dimensional irreducible complex representation is one-dimensional. A continuous character lifts along $t\mapsto e^{it}$ to a continuous homomorphism $\mathbb R\to\mathbb C^\times$, hence has the form $t\mapsto e^{i\lambda t}$. Periodicity by $2\pi$ forces $\lambda=n\in\mathbb Z$, giving precisely $z\mapsto z^n$.