Is R with lower limit topology normal?
$\displaystyle \mathbb{Re}_l$ is regular and Lindelof, so it is normal.
Let X be $\displaystyle \mathbb{Re}$ with lower limit topology.
If $\displaystyle a \in X$ and C is a closed subset of X not containing a, then there is a basic open set [a, b) contained in $\displaystyle X\setminus C$. Then [a, b) and $\displaystyle X \setminus [a,b) $ are disjoint open sets containing a and C, respectively. Thus X is regular.
$\displaystyle \mathbb{Re}_l$ is also Lindelof. It is explained in detail in Munkres p 192.
However, $\displaystyle \mathbb{Re}_l \times \mathbb{Re}_l$ is neither Lindelof nor normal (see here).