Definition:Open Neighborhood
Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$ be a subset of $S$.
Let $N_A$ be a neighborhood of $A$.
Let:
- $N_A \in \tau$
That is, let $N_A$ itself be an open set of $T$.
Then $N_A$ is called an open neighborhood of $A$ in $T$.
Open Neighborhood of Point
Let $x \in S$ be a point of $S$.
Let $N_x$ be a neighborhood of $x$ in $T$.
Let:
- $N_x \in \tau$
That is, let $N_x$ itself be an open set of $T$.
Then $N_x$ is called an open neighborhood of $x$ in $T$.
Real Analysis
Real Numbers
Let $x\in\R$ be a real number.
Let $I \subseteq \R$ be a subset.
Then $I$ is an open neighborhood of $x$ if and only if $I$ is open and $I$ is a neighborhood of $x$.
Real Euclidean Space
Let $n\geq1$ be a natural number.
Let $x\in\R^n$.
Let $I \subseteq \R$ be a subset.
Then $I$ is an open neighborhood of $x$ if and only if $I$ is open and $I$ is a neighborhood of $x$.
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
- Set is Open iff Neighborhood of all its Points
- Open Superset is Open Neighborhood: $N_A$ is an open neighborhood of $A$ if and only if $N_A$ is an open set in $T$ such that $A \subseteq N_A$.
Linguistic Note
The UK English spelling of neighborhood is neighbourhood.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction