Example of Non-separable stochastic process. This question is related to the link:

https://www.encyclopediaofmath.org/index.php/Separable_process

The link provided a basic definition of separable Stochastic process. I felt all the process under study seems to be separable. Is it true? Is there any example of  Non-separable stochastic process which is of some importance?
Let's consider the following setting: $([0,1],\mathscr L, \mu_L)$ with Lebesgue Set $\mathscr L$ and $t\in[0,1]$. Define the stochastic process as$Y_t=\mathbb 1_{\{\omega\}}(t)$, where $\{\omega\}$ is a singleton. This process just take value 1 at $t=\omega$.(for example one possible sample path is: For $\omega=1$. $Y_t(\omega)=1$ if $t=1$ and $Y_t(\omega)=0$ if $t \neq 1$).

Now we want to show it is not separable:

Consider the close set $A=\{1\}$, the set $\{\omega\in\Omega,Y_t(\omega)\in \{1\}\}$ becomes $\{\omega \in \Omega ,\omega=t\}$ and the union $ \cup_{t\in T\cap O}\{\omega\in\Omega,Y_t(\omega)\in \{1\}\}$ for some open interval $O$ is just the set ${T\cap O}$. Similar for a countable set $S_1,S_2,...\subset T$. The union $ \cup_{t\in (S_j)_{j\in\mathbb N} \cap O}\{\omega\in\Omega,Y_t(\omega)\in \{1\}\}$ is just $(S_j)_{j\in\mathbb N}\cap O$. The difference between the two set is not a subset of null set.

The advantage that we are dealing with separable process is that we can operate with countable operations (Union and common sigma-Algebra operations). This simplify our problem for stochastic process in continous time a lot. There is a theorem by Loeve, that under certain condition one can always find a equivalent seperable process.
Any process that is right continuous and has left limits is separable,

the poisson jump process is also separable. Just notice that you can know the value of $X(t,\omega)$ by looking at a sequence of rational numbers $q_n \downarrow t$.

To get a non separable process you need to consider a process that has more irregularity, say that is not right continuous. This might occur if you have a subset that is considered Bad and you tele-transport it to infinity as it enters the Bad region. This often happens when we are studying meta-stability, but still you can modify your process (ignoring the times you spent in the bad region - this is called the trace of the process).

You could also find a non separable process if it took value on a non separable space (it can't be a real valued process )

One often deals with separable process since it is a good property when we are dealing with denumerable set operations (as it is the case for a $\sigma$-algebra).