  We propose and study a new approach to the topologization of spaces of
(possibly not all) future-directed causal curves in a stably causal spacetime.
It relies on parametrizing the curves "in accordance" with a chosen time
function. Thus obtained topological spaces of causal curves are separable and
completely metrizable, i.e. Polish. The latter property renders them
particularly useful in the optimal transport theory. To illustrate this fact,
we explore the notion of a causal time-evolution of measures in globally
hyperbolic spacetimes and discuss its physical interpretation.
