Definition:Separated by Closed Neighborhoods/Sets
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B \subseteq S$ such that:
- $\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A^- \cap N_B^- = \O$
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.