  In this paper we characterize [strict] o-boundedness of the free (abelian)
topological group F(X) (A(X)) as well as the free locally-convex linear
topological space L(X) in terms of properties of a Tychonoff space X. These
properties appear to be close to so-called selection principles, which permits
us to show, that (it is consistent with ZFC that) the property of Hurewicz
(Menger) is l-invariant. This gives a method of construction of OF-undetermined
topological groups with strong combinatorial properties.
