  We show that the proper interpretation of the cocycle operators appearing in
the physical vertex operators of compactified strings is that the closed string
target is non-commutative. We track down the appearance of this
non-commutativity to the Polyakov action of the flat closed string in the
presence of translational monodromies (i.e., windings). In view of the
unexpected nature of this result, we present detailed calculations from a
variety of points of view, including a careful understanding of the
consequences of mutual locality in the vertex operator algebra, as well as a
detailed analysis of the symplectic structure of the Polyakov string. We also
underscore why this non-commutativity was not emphasized previously in the
existing literature. This non-commutativity can be thought of as a central
extension of the zero-mode operator algebra, an effect set by the string length
scale -- it is present even in trivial backgrounds. Clearly, this result
indicates that the $\alpha'\to 0$ limit is more subtle than usually assumed.
