# 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.

## Linguistic Note

The UK English spelling of neighborhood is neighbourhood.