Definition:Separated by Closed Neighborhoods

Definition
Let $\left({X, \vartheta}\right)$ be a topological space.

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

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

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

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

Points
Let $x, y \in X$ such that:


 * $\exists N_x, N_y \subseteq X: \exists U, V \in \vartheta: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \varnothing$

where $N_x^-$ and $N_y^-$ are the closures of $N_x$ and $N_y$ respectively.

That is, that $x$ and $y$ both have neighborhoods in $X$ whose closures are disjoint.

Then $x$ and $y$ are described as separated by closed neighborhoods.

Equivalence of Definitions
It is clear that separation by closed neighborhoods of two points $x$ and $y$ is the same as for the two singleton sets $\left\{{x}\right\}$ and $\left\{{y}\right\}$.