For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in such a way, that:

i) The map $S\rightarrow \mathbb{Z}[S]$ is a homeomorphism onto the image.

ii) The addition and the inverse map are continuous

And if it is possible, is this topology unique?