Definition:Neighborhood

Topology

Let $\left({X, \vartheta}\right)$ be a topological space.

Neighborhood of a Set

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

A neighborhood of $A$, which can be denoted $N_A$, is any subset of $X$ containing an open set which itself contains $A$.

That is:

$\exists U \in \vartheta: A \subseteq U \subseteq N_A \subseteq X$

Neighborhood of a Point

The set $A$ can be a singleton, in which case the definition is of the neighborhood of a point.

Let $z \in X$ be a point in a $X$.

A neighborhood of $z$, which can be denoted $N_z$, is any subset of $X$ containing an open set which itself contains $z$.

That is:

$\exists U \in \vartheta: z \in U \subseteq N_z \subseteq X$

Open Neighborhood

If $N_A \in \vartheta$, i.e. if $N_A$ is itself open in $X$, then $N_A$ is called an open neighborhood.

Some authorities require all neighborhoods to be open.

Closed Neighborhood

If $\complement_X \left({N_A}\right) \in \vartheta$, i.e. if $N_A$ is closed in $X$, then $N_A$ is called a closed neighborhood.

Metric Space

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

Let $a \in A$.

Let $\epsilon \in \R: \epsilon > 0$ be a positive real number.

The $\epsilon$-neighborhood of $a$ in $M$ is defined as:

$N_\epsilon \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$

If it is necessary to show the metric itself, then the notation $N_\epsilon \left({a; d}\right)$ can be used.

From the definition of open set in the context of metric spaces, it follows that an $\epsilon$-neighborhood in a metric space $M$ is open in $M$.

Neighborhood in Pseudometric Space

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

The $\epsilon$-neighborhood of $a$ in $M$ is defined in exactly the same way as for a metric space:

$N_\epsilon \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$

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 $\epsilon \in \R: \epsilon > 0$ be a positive real number.

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

$N_\epsilon \left({z_0}\right) := \left\{{z \in \C: \left|{z - z_0}\right| < \epsilon}\right\}$

In this context, a neighborhood is often referred to as an open disk (UK spelling: open disc).

Real Analysis

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

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

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

Linguistic Note

The UK English spelling of this is neighbourhood.