Definition:Neighborhood (Topology)

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

Neighborhood of a Point
The set $A$ can be a singleton, in which case the definition is of the neighborhood of a point.

Also defined as
Some authorities define a neighborhood as what is defined on this site as an Open Neighborhood.

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.

Also see

 * From this definition, it follows directly that $X$ itself is always a neighborhood of any $A \subseteq X$.


 * It also follows that any open set of $X$ containing $A$ is a neighborhood of $A$.


 * Neighborhood of All its Points is Open

Linguistic Note
The UK English spelling of this is neighbourhood.