Definition:Deleted Neighborhood

Definition

Real Analysis

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

Let $\map {N_\epsilon} \alpha$ be the $\epsilon$-neighborhood of $\alpha$:

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

Then the deleted $\epsilon$-neighborhood of $\alpha$ is defined as:

$\map {N_\epsilon} \alpha \setminus \set \alpha$.

That is, it is the $\epsilon$-neighborhood of $\alpha$ with $\alpha$ itself removed.

It can also be defined as:

$\map {N_\epsilon} \alpha \setminus \set \alpha : = \set {x \in \R: 0 < \size {\alpha - x} < \epsilon}$

or

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

from the definition of $\epsilon$-neighborhood.

Complex Analysis

Let $z_0 \in \C$ be a point in the complex plane.

Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$.

Then the deleted $\epsilon$-neighborhood of $z_0$ is defined as $\map {N_\epsilon} {z_0} \setminus \set {z_0}$.

That is, it is the $\epsilon$-neighborhood of $z_0$ with $z_0$ itself removed.

It can also be defined as:

$\map {N_\epsilon} {z_0} \setminus \set {z_0} : = \set {z \in A: 0 < \cmod {z_0 - z} < \epsilon}$

from the definition of $\epsilon$-neighborhood.

Metric Space

Let $M = \struct {A, d}$ be a metric space.

Let $x \in A$.

Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball neighborhood of $x$.

Then the deleted $\epsilon$-neighborhood of $x$ is defined as $\map {B_\epsilon} x \setminus \set x$.

That is, it is the open $\epsilon$-ball neighborhood of $x$ with $x$ itself removed.

It can also be defined as:

$\set {y \in A: 0 < \map d {x, y} < \epsilon}$

Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $x \in X$.

Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball neighborhood of $x$.

Then the deleted $\epsilon$-neighborhood of $x$ is defined as $\map {B_\epsilon} x \setminus \set x$.

That is, it is the open $\epsilon$-ball neighborhood of $x$ with $x$ itself removed.

It can also be defined as:

$\set {y \in X: 0 < \norm {x - y} < \epsilon}$

Topology

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Let $V \subseteq S$ be a neighborhood of $x$.

Then $V \setminus \set x$ is called a deleted neighborhood of $x$.

That is, it is a neighborhood of $x$ with $x$ itself removed.

Also known as

A deleted neighborhood is also called a punctured neighborhood.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): deleted neighbourhood