About perfectly normal spaces. In some books the terms: regular and $T_3$; normal and $T_4$; completely normal and $T_5$; perfectly normal and $T_6$ are synonyms, but in some books, the difference is that regular, normal, completely regular and perfectly regular spaces are not $T_1$ and $T_3, T_4, T_5, T_6$ spaces are $T_1$. 
If we take the second definitions (that regular and $T_3$ and other pairs are not synonymous), than $T_6 \implies T_5 \implies T_4\implies T_3$, but completely normal $\implies$ normal but normal spaces do not include regular spaces. I know that perfectly normal $\implies$ completely normal.
But I am interested in: Does there exist 


*

*A perfectly normal space that is $T_0$ but not $T_1$. 

*A perfectly normal space that is not $T_0$ and is not regular.  

*A perfectly normal space that is not $T_0$, is regular but is not completely regular. 

*A perfectly normal space that is not $T_0$ and is completely regular.


There are no examples for (2) and (3), for essentially the same reason that (as proved in Henno's answer) there are no examples for (1).  A perfectly normal space must be $ \mathrm R _ 0 $ (which is a non-$ \mathrm T _ 0 $ version of $ \mathrm T _ 1 $), and Henno's answer is just the $ \mathrm T _ 0 $ version of the proof of that; a direct proof can be found at the self-answered question Perfectly normal spaces are completely regular by @PatrickR (which is what brought this to my attention).  And it's well-known that an $ \mathrm R _ 0 $ normal space must be completely regular (by an argument that's also pretty much contained in Henno's answer).
On the other hand, Henno's answer already has an example of (4): any nontrivial indiscrete space.
