On free groups in classes of groups with topologies

We study properties of free groups in distinct classes of groups with topologies. The conditions under which the quasi-metric on the space of generators $X$ is extended to an invariant quasi-metric on a free group $F(X,\mathcal V)$ in the fixed quasi-variety $\mathcal V$ of groups with topologies are given. This result is applied to the study:
– of free paratopological groups;
– of free quasitopological groups;
– of free semitopological groups;
– of free left topological groups.