Definition:Open Neighborhood

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

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

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

If $N_A \in \tau$, i.e. if $N_A$ is itself open in $T$, then $N_A$ is called an open neighborhood.

Linguistic Note
The UK English spelling of this is neighbourhood.

Also see

 * Set is Open iff Neighborhood of all its Points
 * Open Superset is Open Neighborhood: $N_A$ is an open neighborhood of $A$ iff $N_A$ is an open set in $T$ such that $A \subseteq N_A$.