1. ## Normal space

Is R with lower limit topology normal?

2. Originally Posted by math.dj
Is R with lower limit topology normal?
$\mathbb{Re}_l$ is regular and Lindelof, so it is normal.

Let X be $\mathbb{Re}$ with lower limit topology.

If $a \in X$ and C is a closed subset of X not containing a, then there is a basic open set [a, b) contained in $X\setminus C$. Then [a, b) and $X \setminus [a,b)$ are disjoint open sets containing a and C, respectively. Thus X is regular.

$\mathbb{Re}_l$ is also Lindelof. It is explained in detail in Munkres p 192.

However, $\mathbb{Re}_l \times \mathbb{Re}_l$ is neither Lindelof nor normal (see here).