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.

From Open Superset is Open Neighborhood, $N_A$ is an open neighborhood of $A$ iff $N_A$ is an open set in $T$ such that $A \subseteq N_A$.

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.

However, this treatment is less common, and considered by many to be old-fashioned.

When the term neighborhood is used on this site, it is assumed to be not necessarily open unless so specified.

Linguistic Note
The UK English spelling of this is neighbourhood.

Also see

 * Open Superset is Open Neighborhood