Prove or disprove the following The topology on real generat

Prove or disprove the following: The topology on real
generated by the subbasis
\"S={[a,b) | a, b in Rationals} union {(a,b] | a, b in Rationals}\"
is equivalent to the discrete topology on the set real.

Solution

[a,b) intersection (c,a] = {a}

=> {a} is open for any a Rational .

We want to show {x} is open for any x irrational

But this is not true

{x} is not open when x is irrational

it cannot be written as a finite intersection of elements of S

intersection of any two elements in S will give either empty set or an open interval (a,b) where both (a,b) are rational or half closed half open intervals with rational endpoints. But no intersection will give {x} with x irrational