Let us begin with a lemma ( see it in the Engelking's book):
If $X$ is a $T_1$ space and for every closed $F$ and every open $W$ that contains $F$ there exists a sequence $W_1$, $W_2$, ... of open subsets of $X$ such that $F\subset \cup_{i}W_i$ and $cl(W_i)\subset W$ for $i=$ 1, 2, ..., then the space $X$ is normal.
My question is this: Are there other certain classes of spaces which can be showed normality by this lemma?
I will give an example as following:
Every second-countable regular space is normal.
Proof: Every regular space with a countable base $B$ satisfies the condition in the lemma, because for any $x\in F$ there is a $U_x \in B$ such that $x \in U_x \subset cl(U_x) \subset W$, the family of the all $U_x$'s is countable and $F \subset \cup_{x\in F}U_x$.