We  prove  that under $CH$  there  exists  a

countable  dense  subset  $X$   of   ${\Bbb R}^{\frak c}$  

such  that ${\Bbb R}^{\frak c}$  is  normal on $X$.  

This  answers  a question of A.V.~Arhangel'ski\v\i.  

Another  result is that there exists a  non-regular

separable Hausdorff  space $Z$ which is normal on  $Y$ 

for everydense  countable $Y \subset Z$.  

It  is  also established  that  there  are 

Tychonoff separable spaces  which  are not  normal 

on any countable  dense subspace.