Free topological groups of spaces and their subspaces

We prove that if X is a Tychonoff topological space, Y a subspace of X, and every bounded continuous pseudometric on Y can be extended to a continuous pseudometric on X, then the free topological group FM(Y) coincides with the topological subgroup of FM(X) generated by Y. For this purpose, a new description for the topology of a free topological group in terms of continuous pseudometrics and group seminorms is given. It follows from what has been shown by Uspenskiı̆ that this result implies the Weil completeness of FM(X) for any Dieudonné complete X. It is also proved that if dim X=0, then ind FM(X)=0.