  In this paper we investigate the topological properties of the space of
differential chains 'B(U) defined on an open subset U of a Riemannian manifold
M. We show that 'B(U) is not generally reflexive, identifying a fundamental
difference between currents and differential chains. We also give several new
brief (though non-constructive) definitions of the space 'B(U), and prove that
it is a separable ultrabornological (DF)-space. Differential chains are closed
under dual versions of fundamental operators of the Cartan calculus on
differential forms. The space has good properties some of which are not
exhibited by currents B'(U) or D'(U). For example, chains supported in finitely
many points are dense in 'B(U) for all open U in M, but not generally in the
strong dual topology of B'(U).
