Examples of algebraic groups that are not topological groups

Question

I understand that an algebraic group may not be a topological group because the continuity of multiplication with respect to the Zariski topology is weaker than that with respect to product topology. But is there a nice example that helps to explain this. Many thanks!

Answer 1

Pick the additive group $(\mathbb{C},+)$ with the Zarisky topology. Then, the map $\mathbb{C}\times\mathbb{C}\to\mathbb{C}$ given by $(a,b)\to a+b$, is not continuous.

To see this, just consider the preimage of $0$. This is just the set of points of the form $(a,-a)$ in $\mathbb{C}\times\mathbb{C}$. This is not closed in $\mathbb{C}\times\mathbb{C}$ with the product topology, since the only closed subsets are either of the form $A\times B\subset \mathbb{C}\times \mathbb{C}$ with $A$ and $B$ finite, or of the form $A\times \mathbb{C}$ with $A$ finite.