Equivalence for Perfectly Normal I'm trying to prove the following equivalence for perfectly normal spaces:


*

*$X$ is perfect and normal

*For every open set $W$ there exists a family $(W_n)_{n \in \omega}$ of open sets such that each $W_n$ is open, $W=\bigcup_n W_n$ and $\overline W_n \subset W$.


I managed to show that 1. implies 2. and that 2. implies that $X$ is perfect, but I need help on showing $X$ is normal.
Edit: $X$ is perfect if and only if every closed subset of $X$ is a $G_\delta$.

Hint. Suppose $X$ satisfies (2), and let $F , E \subseteq X$ be disjoint open sets.  Then there are families $\{ U_n \}_{n \in \omega}$, $\{ V_n \}_{n \in \omega}$ of open subsets of $X$ such that:


*

*$F \subseteq X \setminus E = \bigcup_{n \in \omega} U_n$, and $\overline{U_n} \subseteq X \setminus E$ for all $n$;

*$E \subseteq X \setminus F = \bigcup_{n \in \omega} V_n$, and $\overline{V_n} \subseteq X \setminus F$ for all $n$.


Consider the following questions:


*

*Given $n \in \omega$, what kind of set is $U_m \setminus \bigcup_{i \leq m} \overline{V_i}$? (and $V_n \setminus \bigcup_{j \leq n} \overline{U_j}$?)

*Give $n,m \in \omega$, what is $( U_m \setminus \bigcup_{i \leq m} \overline{V_i} ) \cap ( V_n \setminus \bigcup_{j \leq n} \overline{U_j} )$?

*Given $n \in \omega$, what is $F \cap ( U_n \setminus \bigcup_{i \leq n} \overline{V_i} )$?  What is $E \cap ( V_n \setminus \bigcup_{i \leq n} \overline{U_i} )$?


Answering these questions should lead you to disjoint open sets $U,V$ such that $F \subseteq U$ and $E \subseteq V$.

For some extra content, this is closely related to the following

Fact. A space $X$ is normal iff for every closed $F \subseteq X$ and every open $U \supseteq F$ there is a family $\{ W_n \}_{n \in \omega}$ of open subsets of $X$ such that $F \subseteq \bigcup_{n \in \omega} W_n$ and $\overline{W_n} \subseteq U$ for each $n \in \omega$.

This fact allows you to prove some useful results:


*

*Every countable regular space is normal.

*Every second-countable regular space is normal.

*Every regular Lindelöf space is normal.

