Regular Lindelöf Space is Normal Space

Theorem

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

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 $T_3$ Lindelöf Space is $T_4$ Space:

$T$ is a $T_4$ space

By definition, $T$ is a normal space.

$\blacksquare$