Why the topological dimension of C is 2?

Question

From what I know, the topological dimension of a set has to do with open sets covering it, homeomorphic to R^{n}. Then we can cover C with balls, for instance,of R^{2}, is that the reason?

Answer 1

I think you are referring to the dimension of $\mathbb{C}$ as a topological manifold. You can cover $\mathbb{C}$ with open discs with of radius 2, where the centres are on the points $a+ib$ for $a,b$ integers, and map each of these discs homeomorphically onto $D=\{(x,y) \in \mathbb{R}^2:x^2+y^2 \leq 2\}$ . Since $D$ is an open subset of $\mathbb{R}^2$, the space $\mathbb{C}$ is a 2-dimensional topological manifold.