Regular Space with Sigma-Locally Finite Basis is Normal Space

Theorem

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

Let $\BB$ be a $\sigma$-locally finite basis.

Then:

$T$ is a normal space

Proof

By definition of regular space:

$T$ is a $T_3$ space

$T$ is a $T_0$ (Kolmogorov) space

From Regular Space is $T_2$ Space:

$T$ is a $T_2$ space

From $T_2$ Space is $T_1$ Space:

$T$ is a $T_1$ space

From $T_3$ Space with Sigma-Locally Finite Basis is $T_4$ Space:

$T$ is a $T_4$ space

By definition, $T$ is a normal space.

$\blacksquare$