Free topological groups of weak P-spaces

It is known that the free topological group over the Tychonoff space X, denoted F(X), is a P-space if and only if X is a P-space. This article is concerned with the question of whether one can characterize when F(X) is a weak P-space, that is, a space where all countable subsets are closed. Our main result is that F(X) is a weak P-space if and only if X is a weak P-space and every countable subset is C-embedded.