Basically it fundamentally comes to the fact that if is the set of all functions from then should be countable. Namely.

there exists a map from .

adding 2 functions becomes adding their evaluation point wise. for all .

basically we can take the element in we attached a label "1" to and evaluate it there for both 2 functions, label "2" and evaluate it there,... for all natural numbers.

Again, we can do multiplication pointwise as well.