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.

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.

In Engelking's General topology, in exercises part, there is Ju. M. Smirnov's charactarization of normal spaces:

T1 space is normal iff following properties hold (both): a. every closed G-delta set is zero-set; b. 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. Than there is written that, neither of the properties by itself implies the normaliry of X. For b, maybe for example may use co-finite topology on countable set, because any closed set is also G-delta (Also, for that example may also used Niemetzki plane by the same argument).

But, I can't find example of T1 space, which has property a. and is not normal. Thanks for any help.

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.