  We introduce the setting of extended metric-topological measure spaces as a
general "Wiener-like" framework for optimal transport problems and nonsmooth
metric analysis in infinite dimension.
  After a brief review of optimal transport tools for general Radon measures,
we discuss the notions of the Cheeger energy, of the Radon measures
concentrated on absolutely continuous curves, and of the induced "dynamic
transport distances." We study their main properties and their links with the
theory of Dirichlet forms and the Bakry-\'Emery curvature condition, in
particular concerning the contractivity properties and the EVI formulation of
the induced Heat semigroup.
