Definition of separable stochastic process. Which is the "intuition" behind such a definition?

Question

Herebelow, I quote Kuo (2006)

Definition: A stochastic process $X(t,\omega)$, $0\leq t\leq 1$, $\omega\in\Omega$ is called separable if there exist $\Omega_0$ with $P(\Omega_0)=1$ and a countable dense subset $S$ of $[0,1]$ such that for any closed set $F\subset\mathbb{R}$ and any open interval $I\subset [0,1]$, the set difference $$\bigg\{\omega\in\Omega; X(t,\omega)\in\ F, \forall \text{ } t\in I\cap S\bigg\}\backslash\bigg\{\omega\in\Omega; X(t,\omega)\in F,\forall \text{ }t\in I\bigg\}$$ is a subset of the complement $\Omega_0^c$ of $\Omega_0$. The set $S$ is called a separating set.

First, isn't $\Omega_0^c=\emptyset$?

Secondly, what does the fact that $S$ is a subset of the complement $\Omega_0^c$ of $\Omega_0$ mean? What does $S$ "separate"? From what?

More generally, could you please explain the intuition behind such a definition, even in very rough terms? What is it useful for?

Answer 1

Separability means that the behavior of the process is essentially determined by its values on a countable ("separating") set. This is actually not a stochastic but rather an analytic notion. One may similarly define separability of a non-random function:

$f\colon A\to B$ is separable, if there exists a countable dense subset $S\subset A$ with the property: for any closed $F\subset B$ and any open $I\subset A$, if $f(t)\in F$ for all $t\in I\cap S$ , then $f(t)\in F$ for all $t\in I$.

And then one may call a process separable if $X(\cdot,\omega)$ is separable for almost all $\omega\in \Omega$ (which is precisely the usual definition).

Concerning the term itself, it originates from "separable set/separable space", where it is also not very suitable: there is nothing being "separated" (see also a relevant discussion here).