Definition:Closed Neighborhood

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

Let $A \subseteq S$ be a subset of $S$.

Let $N_A$ be a neighborhood of $A$.

If $\complement_S \left({N_A}\right) \in \tau$, that is if $N_A$ is closed in $S$, then $N_A$ is called a closed neighborhood.

Linguistic Note
The UK English spelling of this is neighbourhood.