  Directed topology is a refinement of standard topology, where spaces may have
non-reversible paths. It has been put forward as a candidate approach to the
analysis of concurrent processes. Recently, a wealth of different frameworks
for, i.e., categories of, directed spaces have been proposed. In the present
work, starting from Grandis's notion of directed space, we propose an
additional condition of saturation for distinguished sets of paths and show how
it allows to rule out exotic examples without any serious collateral damage.
Our saturation condition is local in a natural sense, and is satisfied by the
directed interval (and the directed circle). Furthermore we show in which sense
it is the strongest condition fulfilling these two basic requirements. Our
saturation condition selects a full, reflective subcategory of Grandis's
category of d-spaces, which is closed under arbitrary limits of d-spaces, has
arbitrary colimits (obtained by saturating the corresponding colimits of
d-spaces), and has nice cylinder and cocylinder constructions. Finally, the
forgetful functor to plain topological spaces has both a right and a left
adjoint.
