Tychonoff Space is Regular, T2 and T1

Theorem
Let $\left({X, \vartheta}\right)$ be a $T_{3 \frac 1 2}$ (Tychonoff) space.

Then $\left({X, \vartheta}\right)$ is also:
 * a regular space
 * a $T_2$ (Hausdorff) space
 * a $T_1$ (Fréchet) space.

Proof
Let $T = \left({X, \vartheta}\right)$ be a $T_{3 \frac 1 2}$ (Tychonoff) space.

From the definition of Tychonoff space:


 * $\left({X, \vartheta}\right)$ is a completely regular space
 * $\left({X, \vartheta}\right)$ is a Kolmogorov ($T_0$) space.

We have that a completely regular 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.