Real Number Line satisfies all Separation Axioms

Theorem
Let $\left({\R, \tau_d}\right)$ be the real number line under the Euclidean metric considered as a space.

Then $\left({\R, \tau_d}\right)$ fulfils all separation axioms:


 * $\left({\R, \tau_d}\right)$ is a $T_0$ (Kolmogorov) space
 * $\left({\R, \tau_d}\right)$ is a $T_1$ (Fréchet) space
 * $\left({\R, \tau_d}\right)$ is a $T_2$ (Hausdorff) space


 * $\left({\R, \tau_d}\right)$ is a semiregular space


 * $\left({\R, \tau_d}\right)$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space


 * $\left({\R, \tau_d}\right)$ is a $T_3$ space
 * $\left({\R, \tau_d}\right)$ is a regular space
 * $\left({\R, \tau_d}\right)$ is an Urysohn space
 * $\left({\R, \tau_d}\right)$ is a $T_{3 \frac 1 2}$ space
 * $\left({\R, \tau_d}\right)$ is a Tychonoff (completely regular) space
 * $\left({\R, \tau_d}\right)$ is a $T_4$ space
 * $\left({\R, \tau_d}\right)$ is a normal space
 * $\left({\R, \tau_d}\right)$ is a $T_5$ space
 * $\left({\R, \tau_d}\right)$ is a completely normal space
 * $\left({\R, \tau_d}\right)$ is a perfectly $T_4$ space
 * $\left({\R, \tau_d}\right)$ is a perfectly normal space

Proof

 * Real Number Line is Metric Space
 * Metric Space fulfils all Separation Axioms