Definition:Neighborhood (Topology)

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This page is about neighborhoods in the context of topology. For other uses, see Definition:Neighborhood.

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:

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


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


Linguistic Note

The UK English spelling of this is neighbourhood.