about non normality of Sorgenfrey plane There is a classical theorem saying that a regular space with a countable basis is normal.  The Sorgenfrey plane is regular since it is the product of two regular spaces (which in fact are normal) and it has a countable basis since it is separable which would say that it should be normal, however, it is NOT.  
What am I thinking wrong? 
Thanks in advance.

Countable basis $\Longrightarrow$ separable | in general topological spaces.
Separable $\Longrightarrow$ countable basis | only in metrizable topological spaces.

Sorgenfrey’s plane is not metrizable. It is separable, but does not admit a countable basis.
