\begin{definition}[Definition:Separated by Neighborhoods/Points/Neighborhoods]
Let $T = \struct {S, \tau}$ be a topological space.
Let $x, y \in S$ such that:
:$\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \in U \subseteq N_x, y \in V \subseteq N_y: N_x \cap N_y = \O$
That is, that $x$ and $y$ both have neighborhoods in $T$ which are disjoint.
Then $x$ and $y$ are described as '''separated by neighborhoods'''.
Thus two points are '''separated by neighborhoods''' $x$ and $y$ {{iff}} the two singleton sets $\set x$ and $\set y$ are separated by neighborhoods as sets.
Category:Definitions/Separated by Neighborhoods
\end{definition}