Image countable when state space is not? From Jacod / Protter: "Probability Essentials", Springer:

Note that even if the state space (or range space) $T$ is not countable, the image $T'$ of $\Omega$ under $X$ (that is, all points $\{i\}$ in $T$ for which there exists an $\omega\in\Omega$ such that $X(\omega) = i$ ) is either finite or countably infinite.

(where $X$ is a function (random variable) from $\Omega$ into a set $T$)
I do not understand this. If $T$ is the uncountable set $\bf R$ (the real numbers), could the image also be uncountably infinite?

Generally speaking, if $f$ is a function then $f$ is always onto its range. If the domain of $f$ is countable (or generally can be well ordered) then $f$ has a right inverse, and therefore $|\operatorname{Rng}(f)|\le|\operatorname{Dom}(f)|$.
By this property we have that if the domain of $f$ is countable then its range is at most countable.
