TITLE: An example of a group with a topology
QUESTION [6 upvotes]: Do you know an example of a group with a topology satisfying both the following two conditions

the product is separately continuous but not jointly continuous
the inversion map is continuous.

REPLY [11 votes]: Take, for instance, $(\mathbb{Q},+)$, endowed with the Zariski topology (that is, a non-empty set $A$ is open if and only if $A^\complement$ is finite). Then the inversion ($x\mapsto-x$) is clearly continuous and addition is clearly separately continuous. But it is not jointly continuous since, for instance $\{(x,y)\in\mathbb{Q}^2\,|\,x+y=0\}$ is not a closed set.

REPLY [7 votes]: Let $G$ be any infinite group and give it the cofinite topology.  Then the product is separately continuous as is inversion, since any bijection $G\to G$ is continuous.  But the product is not jointly continuous, since $\{1\}$ is closed but its preimage is not.  (Or, you can just cite the fact that any $T_0$ topological group is Hausdorff, so $G$ cannot be a topological group.)