In Engelking's *General topology*, in the exercises section, there is Ju. M. Smirnov's characterization of normal spaces:


A $T\_1$ space is normal iff the following properties hold (both):


1. Every closed $G\_\delta$ set is zero-set;
2. for every $F$ closed set and $G$ open set, such that $F$ is in $G$, there exist $M$ closed $G\_\delta$ set, such that $F$ is in $M$ and $M$ is in $G$.


This equivalency is not hard to prove. Then there is written that, neither of the properties by itself imply the normality of $X$. For 2, maybe for example one may use the co-finite topology on a countable set, because any closed set is also $G\_\delta$ (Also, for that example one may use Niemetzki plane by the same argument).


But I can't find an example of $T\_1$ space, which has the property 1 and is not normal. Thanks for any help.



An example of a $T_1$ space where all closed $G_\delta$ sets are zero-sets, but it isn't normal
Observe that every function $f:\omega\_{1}\rightarrow\mathbb{R}$ is eventually constant and $\omega\_{1}$ is normal. Observe also that if $A\subseteq\omega\_{1}$ is a closed $G\_{\delta}$ set, then the characteristic function $\chi\_{A}$ of $A$ is eventually constant.


Let $X=((\omega\_{1}+1)\times(\omega+1))\setminus\{(\omega\_{1},\omega)\}$. $X$ is known as the Tychonoff plank. $X$ is a typical example of a completely regular space that is not normal since the closed sets $\omega\_{1}\times\{\omega\}$ and
$\{\omega\_{1}\}\times\omega$ cannot be separated by an open set.


Now, if $A$ is a closed $G\_{\delta}$ subset of $X$, then $X\cap(\omega\_{1}+1)\times\{n\}$ is eventually constant for each $n$. Therefore, there is a subset $D\subseteq\omega+1$, and an ordinal $\alpha<\omega\_{1}$ and a closed subset
$C\subseteq \omega+1$ such that $A=D\cup(\alpha,\omega+1]\times C\setminus\{(\omega\_{1},\omega)\}$. One can show that $A$ is a zero set of some continuous function $f:X\rightarrow[0,1]$. Thus, every closed $G\_{\delta}$ set in $X$ is a zero set.


