Question

show that the countable collection B={(a,b):a<b,a and b rational} is the basis that generates the standard topology on R

Answer 1

Given that

B = { (a, b)| a < b, a, b ∈ Q }.

Note that the standard topology on R is generated by open intervals (a, b) where a, b ∈ R. So by definition of a topology generated by a basis, for any open set U and any x ∈ U,  

we have x ∈ (a, b) ⊂ U,

for some a, b ∈ R. Now, by density of Q in R, there exist c, d ∈ Q such that a ≤ c < x and x < d ≤ b, or,
in other words:
x ∈ (c, d) ⊂ (a, b) ⊂ U and c, d ∈ Q.

Since by Lemma

(X be a topological space. Suppose that C is a collection of open sets of X such that for each open set U of X and each x in U, there is an element C of C such that x ∈ C ⊂ U.Then C is a basis for the topology of X).

the above B is a basis for the standard topology on R.