One proof of a version of Cauchy's theorem actually uses the fact that this is not the case. You take the intersection of some specific nested triangles in the complex plane, which shrink to exactly one point. It seems like you had countably infinite sets in mind, but this proof is too pretty...
I think this really emphasizes the role the sigma algebra plays in probability. It is viable to say the probability of selecting a multiple of 3 is 1/3, so long as you restrict the measurable events appropriately. You could restrict your measurable events to the sigma algebra generated by the...