Regular Space is Semiregular Space

Theorem
Let $\left({S, \tau}\right)$ be a regular space.

Then $\left({S, \tau}\right)$ is also a semiregular space.

Proof
Let $T = \left({S, \tau}\right)$ be a regular space.

From the definition:


 * $\left({S, \tau}\right)$ is a $T_3$ space
 * $\left({S, \tau}\right)$ is a $T_0$ (Kolmogorov) space.

We also have that a $T_3$ Space is Semiregular.

Hence the result, by definition of semiregular space.