nLab complex n-dimensional space

Given a positive integer $n$, the complex $n$-dimensional space $\mathbb{C}^n$ is the set of $n$-tuples of complex numbers considered as a $n$-dimensional topological vector over the field of complex numbers $\mathbb{C}$ (such a Hausdorff topology is unique). For a continuous map from a complex $n$-dimensional space to a complex $m$-dimensional space this structure is enough to define complex differentiability of a map, hence the complex analytic structure is induced.