Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point

Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.

Let $p$ be a new element not in $\Q$.

Let $\Q^* := \Q \cup \set p$.

Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\left({\Q, \tau_d}\right)$.

Then $p$ is a dispersion point of $T^*$.

Proof
By definition, $p$ is a dispersion point of $T^*$ :
 * $\Q^*$ is a connected set in $T^*$
 * $\Q^* \setminus \set p$ is totally disconnected in $T^*$.

From Alexandroff Extension of Rational Number Space is Connected, $\Q^*$ is a connected set in $T^*$.

It remains to be shown that $\Q^* \setminus \set p$ is totally disconnected in $T^*$.

By definition, $\Q^* \setminus \set p$ is the rational number space $\struct {\Q, \tau_d}$.

The result follows from Rational Numbers are Totally Disconnected.