\begin{definition}[Definition:Indiscrete Extension of Reals]
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$.
Let $\BB$ be the set of sets:
:$\BB := \set {H: \exists U \in \tau_d: H = U \cap D}$
Let $\tau^*$ be the topology generated from $\tau_d$ by the addition of all sets of $\BB$.
:$\tau^* = \tau_d \cup \BB$
$\tau^*$ is then referred to as an '''indiscrete extension of $\R$'''.
It is usual to focus attention on the two specific cases where $D$ is either the set of rational numbers or the set of irrational numbers:
\end{definition}