# Real Number Line satisfies all Separation Axioms

Jump to navigation Jump to search

## Theorem

Let $\struct {\R, \tau_d}$ be the the real number line with the usual (Euclidean) topology.

Then $\struct {\R, \tau_d}$ fulfils all separation axioms:

$\struct {\R, \tau_d}$ is a $T_0$ (Kolmogorov) space
$\struct {\R, \tau_d}$ is a $T_1$ (Fréchet) space
$\struct {\R, \tau_d}$ is a $T_2$ (Hausdorff) space
$\struct {\R, \tau_d}$ is a semiregular space
$\struct {\R, \tau_d}$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space
$\struct {\R, \tau_d}$ is a $T_3$ space
$\struct {\R, \tau_d}$ is a regular space
$\struct {\R, \tau_d}$ is an Urysohn space
$\struct {\R, \tau_d}$ is a $T_{3 \frac 1 2}$ space
$\struct {\R, \tau_d}$ is a Tychonoff (completely regular) space
$\struct {\R, \tau_d}$ is a $T_4$ space
$\struct {\R, \tau_d}$ is a normal space
$\struct {\R, \tau_d}$ is a $T_5$ space
$\struct {\R, \tau_d}$ is a completely normal space
$\struct {\R, \tau_d}$ is a perfectly $T_4$ space
$\struct {\R, \tau_d}$ is a perfectly normal space

## Proof

$\blacksquare$