  A number of researchers have introduced topological structures on the set of
laws of stochastic processes. A unifying goal of these authors is to strengthen
the usual weak topology in order to adequately capture the temporal structure
of stochastic processes.
  Aldous defines an extended weak topology based on the weak convergence of
prediction processes. In the economic literature, Hellwig introduced the
information topology to study the stability of equilibrium problems. Bion-Nadal
and Talay introduce a version of the Wasserstein distance between the laws of
diffusion processes. Pflug and Pichler consider the nested distance (and the
weak nested topology) to obtain continuity of stochastic multistage programming
problems. These distances can be seen as a symmetrization of Lassalle's causal
transport problem, but there are also further natural ways to derive a topology
from causal transport.
  Our main result is that all of these seemingly independent approaches define
the same topology in finite discrete time. Moreover we show that this 'weak
adapted topology' is characterized as the coarsest topology that guarantees
continuity of optimal stopping problems for continuous bounded reward
functions.
