Another way to make it countable would be to instead go to the category of posets, Then the rational interval basis is a poset with a countable number of elements, and by the Alexandroff construction corresponds to the real line (or at least something very similar). But, this construction gives a full and faithful embedding of the category of posets to the category of spaces (which basically means you get all and only continuous maps from monotonic function).
I guess the ontology version in this case would be the category of prosets. (Personally, I’m not sure that ontology of the universe isn’t a type error).
Another way to make it countable would be to instead go to the category of posets, Then the rational interval basis is a poset with a countable number of elements, and by the Alexandroff construction corresponds to the real line (or at least something very similar). But, this construction gives a full and faithful embedding of the category of posets to the category of spaces (which basically means you get all and only continuous maps from monotonic function).
I guess the ontology version in this case would be the category of prosets. (Personally, I’m not sure that ontology of the universe isn’t a type error).