\begin{definition}[Definition:Indiscrete Extension of Reals/Rational]
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\Q$ denote the set of rational numbers.
Let $\tau^*$ be the indiscrete extension of $\struct {\R, \tau_d}$:
:$\tau^* = \tau_d \cup \set {H: \exists U \in \tau_d: H = U \cap \Q}$
$\tau^*$ is then referred to as the '''indiscrete rational extension of $\R$'''.
\end{definition}