Definition:Neighborhood (Topology)

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

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

Also see

 * Definition:Open Neighborhood
 * Definition:Closed Neighborhood


 * Topological Space is Open Neighborhood of Subset


 * Open Superset is Open Neighborhood


 * Set is Open iff Neighborhood of all its Points

Linguistic Note
The UK English spelling of this is neighbourhood.