Definition:Separated by Closed Neighborhoods/Sets

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A, B \subseteq S$ such that:
 * $\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A^- \cap N_B^- = \varnothing$

where $N_A^-$ and $N_B^-$ are the closures in $T$ of $N_A$ and $N_B$ respectively.

That is, that $A$ and $B$ both have neighborhoods in $T$ whose closures are disjoint.

Then $A$ and $B$ are described as separated by closed neighborhoods.