Definition:Rational Sequence Topology

Definition
Let $\struct {\R, \tau_d}$ denote the real number line with the usual (Euclidean) topology.

Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers

For each $x \in \Bbb I$, let $\sequence {x_i}$ be a sequence of rational numbers which converges to $x$ in $\tau_d$.

Let $\tau$ be the topology defined on $\R$ as:
 * $(1): \quad$ All rational numbers are open points in $\R$


 * $(2): \quad$ The sets $U_n$ of the form:
 * $\map {U_n} x := \sequence {x_i}_n^\infty \cup \set x$
 * form a basis for the irrational point $x$.

$\tau$ is then referred to as the rational sequence topology on $\R$.

Also see

 * Rational Sequence Topology is Topology