Alexandroff Extension of Rational Number Space is Biconnected

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 $\struct {\Q, \tau_d}$.

Then $T^*$ is a biconnected space.

Proof

From Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point, $p$ is a dispersion point of $T^*$.

The result follows from Set with Dispersion Point is Biconnected.

$\blacksquare$

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: $5$