Definition:Open Neighborhood

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

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

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

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

Also defined as
Some authorities define a neighborhood as being open.

That is, in order to be a neighborhood of $A$, $N_A$ must be an open set.

Linguistic Note
The UK English spelling of this is neighbourhood.