This question was given by my instructor to work by myself and I was unable to completely prove it.
(a) Prove that a $T_1$ space (X,T) is normal iff for each closed subset C of X and each open set U such that $C\subseteq U$ , there is an open set V such that $C\subseteq V$ and $\overline{V}\subseteq U$.
(b) Prove that a $T_1$ space (X,T) is normal iff for each pair of disjoint closed subsets C and D of X, there are open sets U and V such that $C\subseteq U$ , $D\subseteq V$ and $\overline{U}\bigcap \overline{V} =\emptyset$
In both parts I have proved the space to be normal assuming the condition given after "iff" but not converse.
In (a) I can always say that $C\subseteq X$ but How to prove the existence of such V?
In (b) It is known that there exists U and V such that $ U \bigcap V = \emptyset $ but how to prove that $\overline{U}\bigcap \overline{V} =\emptyset$
The "hard" part here is proving $(a)$. If you prove it then $(b)$ becomes easy. Indeed, assume that $X$ is normal and let $C,D$ be disjoint closed subsets of $X$. Since $X$ is normal there exist open sets $U_1$ and $V_1$ such that $C \subseteq U_1$, $Y \subseteq V_1$ and $U_1 \cap V_1 = \emptyset$. Now from part $(a)$ we can select open sets $U$ and $V$ such that $C \subseteq U \subseteq \overline{U} \subseteq U_1$ and $D \subseteq V \subseteq \overline{V} \subseteq V_1$. Now verify that $U$ and $V$ satisfy the condition required on $(b)$
Now to prove the forward direction of $(a)$ assume that $X$ is normal and let $C$ be a closed subset of $X$ contained in some open set $U$. Take $B = X \setminus U$. $B$ is closed since $U$ is open. Now we can select disjoint open sets $V, W$ such that $C \subseteq V$ and $B \subseteq W$. Can you see now why $\overline{V} \subseteq U$ (or $\overline{V} \cap B = \emptyset$)?