Regular Lindelöf Space is Normal Space
Jump to navigation
Jump to search
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.
$\blacksquare$