I have a question about $T_6$ (perfectly normal) spaces. I first want to prove that the $T_6$ property is hereditary. I only know the Lemma from Urysohn and Tietze. Then i also want to conclude that every $T_6$ space is also a $T_5$ (completely normal) space, thus that every subspace is normal. Can someone help me?! Thank you for help :)
For my own work: a topologic space is perfectly normal if for every $A,B\subset X$ closed and disjoint, there exists a function $f\in C(X,[0,1])$ such that $f^{-1}(0)=A$ and $f^{-1}(1)=B$, but i can not see how to conlcude that than every subspace is normal.