All abelian groups are top.groups?

Question

I think yes, because all abelian groups isomorphic to some quotient group of free group. And we can represent this quitent group like the Cayley graph, and make the standart metric on the set of vertex, were all edges have the lenght $=c$ ($c$ -- some constant in $\mathbb{R}$). And create the metric topology after that.

How right am I? I am not sure, that all abelian groups can represent by Cayley graph...

Answer 1

Any group becomes a topological group when equipped with the discrete topology (= all sets are open, and hence all functions from it are continuous). So every group is a topological group, abelian or not (or rather: every group is the underlying group of some topological group).