# Tychonoff Space is Regular, T2 and T1

## Theorem

Let $\struct {S, \tau}$ be a Tychonoff space.

Then $\struct {S, \tau}$ is also:

a regular space
a $T_2$ (Hausdorff) space
a $T_1$ (Fréchet) space.

## Proof

Let $T = \struct {S, \tau}$ be a Tychonoff space.

From the definition of Tychonoff space:

$\struct {S, \tau}$ is a $T_{3 \frac 1 2}$ space
$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.

We have that a $T_{3 \frac 1 2}$ space is a $T_3$ space.

From the definition, a regular space is:

a $T_3$ space
a $T_0$ (Kolmogorov) space.

So a Tychonoff space is a regular space.

Then we have a regular space is a $T_2$ (Hausdorff) space.

Then we have a $T_2$ (Hausdorff) space is a $T_1$ (Fréchet) space.

$\blacksquare$