\begin{definition}[Definition:Discrete Extension of Reals/Irrational]
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers.
Let $\BB$ be the set of sets defined as:
:$\BB = \tau_d \cup \set {\set x: x \in \Bbb I}$
Let $\tau*$ be the topology generated from $\BB$.
$\tau^*$ is referred to as the '''discrete irrational extension of $\R$'''.
\end{definition}