\begin{definition}[Definition:Pointed 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 defined as:
:$\BB = \set {\set x \cup \paren {U \cap D}: x \in U \in \tau_d}$
Let $\tau'$ be the topology generated from $\BB$.
$\tau'$ is referred to as a '''pointed 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}