Definition:Neighborhood (Topology)
This page is about Neighborhood in the context of topology. For other uses, see Neighborhood.
Definition
Let $T = \struct {S, \tau}$ 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 neighborhood:
- $N_A$ 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 neighborhood is neighbourhood.