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