# 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