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 $S$.

Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$.


Then $N_z$ is a neighborhood of $z$.


That is:

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


Neighborhood defined as Open

Some authorities define a neighborhood of a set $A$ as what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an open neighborhod:

$N_A$ is a neighborhood of $A$ if and only if $N_A$ is an open set of $T$ which itself contains $A$.

That is, in order to be a neighborhood of $A$ in $T$, $N_A$ must not only be a subset of $T$, but also be an open set of $T$.

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

  • Results about neighborhoods can be found here.


Linguistic Note

The UK English spelling of this is neighbourhood.