Definition:Separated by Closed Neighborhoods

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Definition

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


Sets

Let $A, B \subseteq S$ such that:

$\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A^- \cap N_B^- = \varnothing$

where $N_A^-$ and $N_B^-$ are the closures in $T$ of $N_A$ and $N_B$ respectively.

That is, that $A$ and $B$ both have neighborhoods in $T$ whose closures are disjoint.


Then $A$ and $B$ are described as separated by closed neighborhoods.


Points

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.


Also see


Weaker conditions