Definition:Separated by Neighborhoods

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

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

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

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

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


 * $\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$

That is, that $x$ and $y$ both have neighborhoods in $T$ which are disjoint.

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

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