Definition:Neighborhood (Topology)

Definition

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

Neighborhood of a Set

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

A neighborhood of $A$, which can be denoted $N_A$, is any subset of $S$ containing an open set of $T$ which itself contains $A$.

That is:

$\exists U \in \tau: A \subseteq U \subseteq N_A \subseteq S$

Neighborhood of a Point

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

Let $z \in S$ be a point in a $S$.

A neighborhood of $z$, which can be denoted $N_z$, is any subset of $S$ containing an open set of $T$ which itself contains $z$.

That is:

$\exists U \in \tau: z \in U \subseteq N_z \subseteq S$

Open Neighborhood

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

Closed Neighborhood

If $\complement_S \left({N_A}\right) \in \tau$, that is if $N_A$ is closed in $S$, then $N_A$ is called a closed neighborhood.

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.

Linguistic Note

The UK English spelling of this is neighbourhood.