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An example of a group with a topology

Do you know an example of a group with a topology satisfying both the following two conditions

2 Answers
2

Take, for instance, $(mathbbQ,+)$, endowed with the Zariski topology (that is, a non-empty set $A$ is open if and only if $A^complement$ is finite). The inversion ($xmapsto-x$) is clearly continuous and that addition is clearly separately continuous. But it is not jointly continuous since, for instance $(x,y)inmathbbQ^2,$ is not a closed set.

Let $G$ be any infinite group and give it the cofinite topology. Then the product is separately continuous as is inversion, since any bijection $Gto G$ 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.)

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