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

Theorem
Let $\left({\Q, \tau_d}\right)$ be the rational number space under the Euclidean topology $\tau_d$.

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

Let $\Q^* := \Q \cup \left\{{p}\right\}$.

Let $T^* = \left({\Q^*, \tau^*}\right)$ 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 $T^*$
 * $\Q^* \setminus \left\{{p}\right\}$ 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 $\Q^* \setminus \left\{{p}\right\}$ is totally disconnected in $T^*$.

By definition, $\Q^* \setminus \left\{{p}\right\}$ is the rational number space $\left({\Q, \tau_d}\right)$.

The result follows from Rational Numbers are Totally Disconnected.