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