Definition:Neighborhood (Analysis)

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This page is about neighborhoods in the context of analysis. For other uses, see Definition:Neighborhood.

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:

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


Also see


Internationalization

Neighborhood is translated:

In Dutch: omgeving
In French: voisinage


Linguistic Note

The UK English spelling of neighborhood is neighbourhood.