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 T2 Space:
 * $T$ is a $T_2$ space

From T2 Space is T1 Space:
 * $T$ is a $T_1$ space

From User:Leigh.Samphier/Topology/T3 Lindelöf Space is T4 Space:
 * $T$ is a $T_4$ space

Hence $T$ is a nomal space by definition.