  We argue that apart from the standard closed and open strings one may
consider a third possibility that we call monodromic strings. The monodromic
string propagating on a target looks like an ordinary open string (a mapping
from a segment to the target) but its space of states is isomorphic to that of
a closed string.
  It is shown that the monodromic strings naturally appear in T-dualizing
closed strings moving on simply connected targets. As a nontrivial topology
changing example we show that the monodromic strings on a compact Poisson-Lie
group are T-dual to the standard closed strings propagating on the noncompact
dual PL group.
