If you're going to continue in math, this is an important fact & example that you should understand well.
Having a countable base comes up often enough that it's given it's own name. "A topological space is second countable if it has a countable base." This example, exploiting that the rationals are dense in the reals to produce a countable base (generalized to ), is the "stock example" of what/why/how a base is countable.
(And the exploitability of countable dense subsets is itself a common/useful enough feature to have it's own name, "separable". Furthermore, these two ideas, separable and second countable, are often considered together, and are equivalent properties for nice spaces.)
Again, if you're going to continue in math, it might be worth investing a little time learning some basic LaTex. Your entire post was kinda difficult to read, but would've been very easy to read had it been done in LaTex.