Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without resorting to metrizability.

Normally, one considers a metric space $M$, a closed time interval $T \subseteq \mathbb R$, and the space of càdlàg functions $D(T,M)$. This space is equipped with the Skorokhod topology, which allows one to wiggle space and time a bit.

In the end, we are concerned with have probability distributions on $D(T,M)$ rather than metric structures, which means the whole point of the topology is to just construct a nice Borel $\sigma$-algebra.

Assumptions: Let $T$ be a first-countable topological space equipped with a continuous partial order, and let $M$ be a first-countable topological space. Define the space of càdlàg functions $D(T,M)$ in the natural way.

Question: Under these relaxed assumptions, is there a natural topology on $D(T,M)$? That is, a topology which coincides with the usual Skorokhod topology in the setting of $T \subseteq \mathbb R$ and $M$ metrizable.


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