I quote Kuo (2006).
Definition A stochastic process $\tilde{X}(t)$ is called a version of $X(t)$ if $\mathbb{P}\{\tilde{X}(t)=X(t)\}=1$ for each $t$.
Definition A stochastic process $\tilde{X}(t)$ is called a realization of $X(t)$ if there exists $\Omega_0$ such that $\mathbb{P}(\Omega_0)=1$ and for each $\omega\in\Omega_0, \tilde{X}(t,\omega)=X(t,\omega)$ for all $t$.
Then, I read that:
Obviously, a realization is also a version, but not vice versa.
and that is pretty clear to me.
What I cannot understand is the following statement:
But if the time parameter set is countable, then these two concepts are equivalent.
Why is the immediately above statament true?
That's because a countable union of null sets is a null set, i.e., if the (time) parameter set $T$ is countable, then $$ \mathsf{P}\!\left(\bigcup_{t\in T}\{\tilde{X}_t\ne X_t\}\right)\le \sum_{t\in T}\mathsf{P}\!\left(\tilde{X}_t\ne X_t\right)=0. $$