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 $T^*$ if and only if:

$\Q^*$ 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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $35$. One Point Compactification Topology: $5$