Discrete random variables are defined on a countable probability space. All examples I found are either finite spaces or something like all integers

Question

I wonder if a discrete random variable could be defined on the set {1, 2, 1+1/2, 1-1/2, 2+1/2, 2-1/2, 1+1/3, 1-1/3, 2+1/3, 2-1/3, 1+1/4, ...} i.e on a countable set with 2 accumulation points (or even only 1). In this example there would be no way to order the jump points by magnitude. Kalbfleisch and Prentice are using this ordering implicitly in their book "The statistical analysis of failure time series" when explaining the probability function in chapter 1.2.2 with the 2 bullets at the end. They don't explicitly restrict the discrete case to sets w/o accumulation points

Answer 1

You absolutely can, and it's pretty simple to do so.

If your support set $A$ is countable, then there exists a bijection $f: \mathbb{N} \rightarrow A$. So then all you need to do is construct a random variable $Y$ on the natural numbers, and then your random variable is just $X = f(Y)$.