Separation Properties of Alexandroff Extension of Rational Number Space

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 $T^*$ satisfies no Tychonoff separation axioms higher than a $T_1$ (Fréchet) space.

Proof

From Alexandroff Extension of Rational Number Space is $T_1$ Space, $T^*$ is a $T_1$ space.

From Alexandroff Extension of Rational Number Space is not Hausdorff, $T^*$ is not a $T_2$ (Hausdorff) space.

From Completely Hausdorff Space is Hausdorff Space, $T^*$ is not a $T_{2 \frac 1 2}$ (completely Hausdorff) space.

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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: $4$