A note on the adapted weak topology in discrete time

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.