Definition:Separated by Neighborhoods/Points/Open Sets

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

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


 * $\exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

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

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

Thus two points $x$ and $y$ are separated by neighborhoods the two singleton sets $\left\{{x}\right\}$ and $\left\{{y}\right\}$ are separated by open neighborhoods as sets.