Weakly complete free topological groups

A topological group G is sequentially complete if it is sequentially closed in any other topological group. We show that for a Tychonoff space X, the free topological group F(X) is sequentially complete iff the free Abelian topological group A(X) is sequentially complete iff X is sequentially closed in βX. Furthermore, the free precompact Abelian group F(X,PA) is sequentially complete iff the space X is sequentially closed in βX. We consider also other forms of weak completeness, namely ω-completeness and k-completeness, introduced analogously by means of the ω-closure and the k-closure. We prove that the groups A(X) and F(X) are ω-complete (k-complete) iff X is ω-closed (k-closed) in the Dieudonné completion μX of X.