The exponential map exp$: \mathbb{C}\rightarrow\mathbb{C}^* = \mathbb{C}\backslash\{0\}$ is open.

Question

This is the first part of Q4 of §1.2 in Dieck's algtop. I'd like to prove this by showing that image of every open subset in the domain is open. But what exactly is the topology with which the multiplicative group $\mathbb{C}^*$ is endowed?