About perfectly normal spaces.

Question

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

1. A perfectly normal space that is $T_0$ but not $T_1$.
2. A perfectly normal space that is not $T_0$ and is not regular.
3. A perfectly normal space that is not $T_0$, is regular but is not completely regular.
4. A perfectly normal space that is not $T_0$ and is completely regular.

Answer 1

Assuming the definitions in my comment: if $X$ is perfectly normal and $T_0$ then let $x \in X$, and suppose it were not closed, so we have $y \in \overline{\{x\}}$ with $y \neq x$. Then $T_0$ tells us we have an open $U$ containing $x$ but not containing $y$ (the other way around is ruled out in this situation). Apply perfect normality and we get a continuous $f: X \to [0,1]$ that $f^{-1}[\{0\}]=U^\complement$, so in particular $f(y)=0$ and $f(x)=p > 0$. But then $f^{-1}[[0,p)]$ is an open neighbourhood of $y$ missing $\{x\}$, contradiction. So $\{x\}$ must be closed and so $X$ is $T_1$ (and hence $T_2$, etc.). So then a space as in 1. does not exist.

The indiscrete space on $\{0,1\}$ is not $T_0$ but is perfectly normal: it is normal for trivial reasons (no non-trivial disjoint closed sets to consider) and it is a $G_\delta$ space. This is an example for 2, and possibly for 4.?

Answer 2

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.