What is the topology on continuous groups?

Question

I was look at this page where it says that

A group having continuous group operations. A continuous group is necessarily infinite, since an infinite group just has to contain an infinite number of elements. But some infinite groups, such as the integers or rationals, are not continuous groups.

If I understand this correctly then with the group operation being a function $G\times G \to G$ there is a topology on $G$ such that $G\times G$ is given the product topology and then the function (group operation) is continuous.

Is that right?

If this is correct, then what is the topology on the group?

My guess would be that one could put any topology on the group, but how does it make sense to say that the integers is not a continuous group?

Answer 1

Turning my comment into an answer:

The linked page appears to be nonsense; I'm surprised to see it at mathworld. There are two possibly-relevant notions whose wikipedia pages I would recommend instead: topological groups (which do not have to be infinite) and Lie groups (which do have to be infinite, if nontrivial).