  We prove that every unconditionally closed subset of a free group is
algebraic, thereby answering affirmatively a 76 years old problem of Markov for
free groups. In modern terminology, this means that Markov and Zariski
topologies coincide in free groups. It follows that the class of groups for
which Markov and Zariski topologies coincide is not closed under taking
quotients. We also show that Markov and Zariski topologies differ from the
so-called precompact Markov topology in non-commutative free groups.
