What results of the form `A sub-X of a free X is free' are known? I am aware of groups, magmas (= set with binary operation), and modules over a principal ideal domain. I'd be interested to hear of structures for which this is either known to be true, or examples where it fails. John Stell There are a lot of counterexamples examples in the topological case. There is also a lot of work on the profinite case (open subgroups of free profinite groups are free profinite). Brown and Hardy proved open subgroups of free topological groups are free topological (various k_{\omega} conditions used) (JLMS (2)10 (1975) 431-440).Sid Morris has a good survey in Categorical Topology (Proceedings Toledo 1983, ed HLBentley et al). I have a paper in Proc AMS 52 (1975) 433-441 giving examples of non projective closed subgroups of the free topological group on [a,b] (e.g. the subgroup generated by {a} and x^2 for x \in [a,b], has \pi_0 = Z_2, which is not free). Ronnie Brown Prof R. Brown Tel: +44 248 382474 School of Mathematics Fax: +44 248 355881 Dean St email: mas010@uk.ac.bangor University of Wales Bangor Gwynedd LL57 1UT UK