Rational sequence topology

For each irrational number x take a sequence of rational numbers {x_k} with the property that {x_k} converges to x with respect to the Euclidean topology.

The rational sequence topology^[1] is specified by letting each rational number singleton to be open, and using as a neighborhood base for each irrational number x, the sets ${\displaystyle U_{n}(x)=\{x_{k}:k\geq n\}\cup \{x\}.}$