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This question is from my topology quiz and I was unable to solve it. So, I am asking for help here.

Prove that $T_1$ -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 $\bar{V} \subseteq U$.

Consider space to be normal , which implies that if there exists disjoint open sets C,D (both closed) then there exists disjoint open sets U,V such that $C\subseteq U $ and $D\subseteq V$. But now, I am unable to understand how to construct V such that the conditions are satisfied. ( One obvious choice of V was U itself but it doesn't works).

For converse, let $C\subseteq U$ and there exists open set V such that $C\subseteq V$ and $\bar{V} \subseteq$ U. Now, let there exists two disjoint closed sets U and V then I have to prove that it is normal . But I can use the assumption given in the question if there exists an open set which I chose X in both cases . Now given is $C\subseteq X $ and $D \subseteq X$ and so there exists V and W such that $\bar{V} \subseteq X$ and $\bar{W} \subseteq X$. But I need to prove that intersection of V and W is empty and I need help with that.

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For the first implication, take $C$ and $X \setminus U$ to be your closed sets and apply the normality condition. This would give you required $V$.

For the other implication: Let $C$ and $D$ be disjoint closed sets. Then, $C \subset X \setminus D$ implies by hypothesis there exists open set $V \subset X \setminus D$ containing $C$. Similarly, you will get an open set $W \subset X \setminus C$ containing $D$. This proves normality.

Ajay Kumar Nair
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