Topology of a free group

An interesting "restriction" is that if you want the topology to be locally compact Hausdorff or come from a complete metric then the topology will have to be the discrete topology. This follow from results in Continuity of homomorphisms by Dudley, where he proves "automatic continuity" results for certain types of groups.

A slight simplification is that if we have $G \to F$ a group homomorphism, where $F$ is free with the discrete topology and $G$ has a nice topology, like being locally compact Hausdorff, then the homomorphism is automatically continuous.

Using this we can suppose that $(F,\tau)$ is one of these nice topologies. Then $$Id:(F, \tau) \to F$$ is continuous by Dudley's automatic continuity results, which means that $\tau$ had to be the discrete topoology.

A consequence of this is that $F_\mathfrak{c}$, the free group on continuum many generators, can not be made into a Polish group. But $F_\mathfrak{c}$ obviously has quotients which can be made Polish, the any countable quotient or the group of real numbers.

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Note that $F_n$ can be given non-locally compact Hausdorff topolgies by embedding $F_n$ into nice groups like Lie groups and be dense in them(related to YCor's comment). These topolgies will "look like" $\mathbb Q$.

With that in mind there are finitely generated groups which have no Hausdorff group topology besides the discrete topology(hence quotients of free groups). The paper On topologizable and non-topologizable groups actually proves that there are lots satisfying even stronger properties.