TITLE: Group with topology which is not topological group
QUESTION [1 upvotes]: What will example of a group $G$ with topology $\tau$ such that $f: G \to G$ such that $f(x) =x^{-1}$ and $g: G \times G \to G$ such that $g((x,y)) = x * y$ (where $*$ is binary operation on $G$) both are not continuous.

REPLY [0 votes]: $(\mathbb R,+)$ with the Zariski topology.
If $g:\mathbb R\times \mathbb R\to \mathbb R$ is given by $g(x,y)=x+y$, then the preimage of, say, $(0,1)$ is the set $-x<y<-x+1$ which is not open for the product of the zariski topologies.
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