  The adapted weak topology is an extension of the weak topology for stochastic
processes designed to adequately capture properties of underlying filtrations.
With the recent work of Bart--Beiglb\"ock-P. as starting point, the purpose of
this note is to recover with topological arguments the intriguing result by
Backhoff-Bartl-Beiglb\"ock-Eder that all adapted topologies in discrete time
coincide. We also derive new characterizations of this topology including
descriptions of its trace on the sets of Markov processes and processes
equipped with their natural filtration. To emphasize the generality of the
argument, we also describe the classical weak topology for measures on $\mathbb
R^d$ by a weak Wasserstein metric based on the theory of weak optimal transport
initiated by Gozlan-Roberto-Samson-Tetali.
