Definition:Neighborhood (Analysis)

This page is about Neighborhoods in the context of Analysis. For other uses, see Neighborhoods.

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:

$\map {N_\epsilon} \alpha := \openint {\alpha - \epsilon} {\alpha + \epsilon}$

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

Topology

Let $T = \struct {S, \tau}$ 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 $M = \struct {A, d}$ 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:

$\map {B_\epsilon} x \subseteq S$

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:

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

Also see

• Definition:Open Neighborhood
• Definition:Closed Neighborhood

• Results about neighborhoods can be found here.

Internationalization

Neighborhood is translated:

In Dutch:	omgeving
In French:	voisinage

Linguistic Note

The UK English spelling of neighborhood is neighbourhood.