Definition 4.10.1 (Normal). Let $X$ be a topological space, then the following are equivalent:
For any $A, B \subset X$ closed with $A \cap B = \emptyset$, there exists $U \in \cn(A)$ and $V \in \cn(B)$ such that $U \cap V = \emptyset$.
For any $A \subset X$ closed and $U \in \cn^{o}(A)$, there exists $V \in \cn^{o}(A)$ such that $A \subset V \subset \ol{V}\subset U$.
Proof. (1) $\Rightarrow$ (2): Since $U^{c}$ is closed, there exists $V \in \cn^{o}(A)$ and $W \in \cn^{o}(U^{c})$ such that $V \cap W = \emptyset$. In which case, $A \subset V \subset \ol{V}\subset W^{c} \subset U$.
(2) $\Rightarrow$ (1): Since $B$ is closed with $A \cap B = \emptyset$, $B^{c} \in \cn^{o}(A)$. Let $V \in \cn^{o}(A)$ with $A \subset V \subset \ol{V}\subset B^{c}$, then $\ol{V}^{c} \in \cn^{o}(B)$ with $V \cap \ol{V}^{c} = \emptyset$.$\square$