Definition:Separated by Closed Neighborhoods/Points

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

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

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

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 $y$ if and only if the two singleton sets $\left\{{x}\right\}$ and $\left\{{y}\right\}$ are separated (as sets) by closed neighborhoods.

Also see