# Definition:Separated by Neighborhoods/Sets

## Definition

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

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

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

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

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