Alexandroff Extension of Rational Number Space is T1 Space
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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 $T_1$ (Fréchet) space.
Proof
From Condition for Alexandroff Extension to be $T_1$ Space, $T^*$ is a $T_1$ space if and only if $\struct {\Q, \tau_d}$ is also a $T_1$ space.
From Rational Numbers form Metric Space, $\struct {\Q, d}$ is a metric space.
From Metric Space fulfils all Separation Axioms, $\struct {\Q, \tau_d}$ is a $T_1$ space.
Hence the result.
$\blacksquare$
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: $4$