\begin{definition}[Definition:Pointed 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:
:$\BB = \set {\set x \cup \paren {U \cap \Bbb I}: x \in U \in \tau_d}$
Let $\tau'$ be the topology generated from $\BB$.
$\tau'$ is referred to as '''pointed irrational extension of $\R$'''.
\end{definition}