I have read an answer of Example of a non extendable rational map, which showed an example of non extendable rational map. I noticed that there is a limit $\lim_{t\to 0}(tx,ty,1)=(0,0,1)$ in affine space $\mathbb{A}^3$, however I confused about how this limit worked? Obviously it could be explained by the limit induced by Zaraski topology, but my question is could it be got by a 'general metric' on affine space $\mathbb{A}^3$ (i.e. The topology induced by this metric is stronger than Zaraski toplogy on $\mathbb{A}^3$ i.e Zaraski topology is a subset of this topology).
An related naive question is does every (algebraically closed) field $\mathbb{K}$ could be endowed with a norm? (Since the metric on $\mathbb{A}^n$ could be induced by the norm on field $\mathbb{K}$)
