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