Answer to Question #114847 in Real Analysis for Sheela John

Question

Give an example of a regular space that is not normal

Answer 1

An example of a regular space that is not normal is the Sorgenfrey plane - $\mathbb{R}_l\times\mathbb{R}_l$   

 It is regular because it is the Cartesian product of regular spaces. It is not normal because any subset 

$A$ of $-\Delta=\{x\times(-x)|x\in\mathbb{R}\}$  is a closed subspace of $\mathbb{R}_l^2$ and it can be shown that there do not exist disjoint open sets about $A$ and $-\Delta \setminus A$ in $\mathbb{R}_l^2$ .