Rational sequence topology

In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set R of real numbers.

Construction

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 is specified by letting each rational number singleton to be open, and using as a neighborhood base for each irrational number x, the sets $U_n(x)=\{x_k:k\ge n\}\cup \{x\}.$