Free topological groups on metrizable spaces and inductive limits

We prove that for a metrizable space X the following are equivalent: (i) the free Abelian topological group A(X) is the inductive limit of the sequence {An(X):n∈N}, where An(X) is formed by all words of reduced length ≤n; (ii) X is locally compact and the set of all non-isolated points of X is separable. In the non-Abelian case, for a metrizable X the following are equivalent: (i) the free topological group F(X) is the inductive limit of the sequence {Fn(X):n∈N}; (ii) X is either locally compact separable or discrete.