\begin{definition}[Definition:Separated by Neighborhoods/Points/Open Sets]
Let $T = \struct {S, \tau}$ be a topological space.
Let $x, y \in S$ such that:
:$\exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$
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''' {{iff}} the two singleton sets $\set x$ and $\set y$ are separated by open neighborhoods as sets.
Category:Definitions/Separated by Neighborhoods
\end{definition}