  We provide some evidence that closed string coordinates will become
non-commutative turning on H-field flux background in closed string
compactifications. This is in analogy to open string non-commutativity on the
world volume of D-branes with B- and F-field background. The class of
3-dimensional backgrounds we are studying are twisted tori (fibrations of a
2-torus over a circle) and the their T-dual H-field, 3-form flux backgrounds
(T-folds). The spatial non-commutativity arises due to the non-trivial
monodromies of the toroidal Kahler resp. complex structure moduli fields, when
going around the closed string along the circle direction. In addition we study
closed string non-commutativity in the context of doubled geometry, where we
argue that in general a non-commutative closed string background is T-dual to a
commutative closed string background and vice versa. Finally, in analogy to
open string boundary conditions, we also argue that closed string momentum and
winding modes define in some sense D-branes in closed string doubled geometry.
