Regular Space is Semiregular Space

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

Then $\struct {S, \tau}$ is also a semiregular space.

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

From the definition:


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

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

Hence the result, by definition of semiregular space.