Alexandroff Extension of Rational Number Space is Biconnected

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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