Real Number Space 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

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28: \ 1$