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).