Is $\mathbb{R}^n$ or $\mathbb{C}^n$ a CW-complex?

Question

I think it's not because it seems that we can't start with zero cells to construct the cw-complex. But how to give a strict proof?

Answer 1

Hint: Here is the structure for $\mathbb R^2$: Put a $0$ cell at the vertices of the standard lattice $\mathbb Z^2$ and fill them with $2$-cells.

Of course the same argument works for $\mathbb C$, and likewise, if you can do it for $\mathbb R^{2n}$ you can do it for $\mathbb C^n$.