Let's say we have a probability space $(\Omega, \sigma, P)$. and a random variable $Y:\Omega\to \mathbb R$
$P$ is of course a mapping $P:\Omega\to \mathbb R$. Now, Omega is not necessarily a topology, but we can simply make the additional assumption that $\Omega$ also has a topology $T$ defined on it. This means that we can talk about whether $Y$ is a continuous function from $\Omega \to \mathbb R$.
The thing is, I usually think of the sample space $\Omega$ as something that "exists in the background", but that is not necessarily what we care about. We really care about the random variables, and when we define probability density functions on them for example, then the sample space is not in view at all (we only see the space of values that the random variable can take (usually $\mathbb R^n$), and the space that the probability density can take (which is $\mathbb R^+$).
So that's why I'm wondering, is there an intuitive interpretation of the random variable being a continuous map from $\Omega$ to $\mathbb R$?
Bonus: If the topology is a differentiable topological space, is there an interpretation of the random variable being a differentiable function?
EDIT: Obviously this will depend on the topology. My question implicitly asks whether there is a topology such that there is such a meaningful interpretation. That is, the interpretation doesn't have to be generally applicable to all topologies and all probability spaces in the same way.