GIVE AN EXAMPLE :
BUT X X Y : not normal
normal is not productive
Sorgenfrey plane .
Even though a Sorgenfrey line is normal, the Sorgenfrey plane is not normal.
Let . As shown by the above link, no disjoint open sets can separate the disjoint closed sets and in . Intuitively, unions of the open rectangles containing and unions of the open rectangles containing in the above link somehow overlaps all the time. A rigorous proof needs more elaboration.
In contrast, and are disjoint closed sets in a Sorgenfrey line , which can be separated by disjoint open sets in (they are both clopen sets).