# Definition:Neighborhood (Analysis)

## Real Analysis

Let $\alpha \in \R$ be a real number.

### Open Subset Neighborhood

Let $N_\alpha$ be a subset of $\R$ which contains (as a subset) an open real set which itself contains (as an element) $\alpha$.

Then $N_\alpha$ is a neighborhood of $\alpha$.

### Epsilon-Neighborhood

On the real number line with the usual metric, the $\epsilon$-neighborhood of $\alpha$ is defined as the open interval:

$N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon \,.\,.\, \alpha + \epsilon}\right)$

where $\epsilon \in \R_{>0}$ is a (strictly) positive real number.

## Topology

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:

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

## Metric Space

Let $M = \left({A, d}\right)$ be a metric space.

Let $S \subseteq A$ be a subset of $A$.

Let $x \in S$.

Let there exist $\epsilon \in \R_{>0}$ such that the open $\epsilon$-ball at $x$ lies completely in $S$, that is:

$B_\epsilon \left({x}\right) \subseteq S$

Then $S$ is a neighborhood of $x$ in $M$.

## Complex Analysis

A specific application of this concept is found in the field of complex analysis:

Let $z_0 \in \C$ be a complex number.

Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

The $\epsilon$-neighborhood of $z_0$ is defined as:

$\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$

## Internationalization

Neighborhood is translated:

 In Dutch: omgeving In French: voisinage

## Linguistic Note

The UK English spelling of neighborhood is neighbourhood.