Definition:Deleted Neighborhood

From ProofWiki
Jump to navigation Jump to search

Definition

Real Analysis

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

Let $N_\epsilon \left({\alpha}\right)$ be the $\epsilon$-neighborhood of $\alpha$:

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


Then the deleted $\epsilon$-neighborhood of $\alpha$ is defined as $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\}$.

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


It can also be defined as:

$N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} : = \left\{{x \in \R: 0 < \left \vert{\alpha - x}\right \vert < \epsilon}\right\}$

or

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

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

Let $x \in A$.

Let $B_\epsilon \left({x}\right)$ be the open $\epsilon$-ball neighborhood of $x$.


Then the deleted $\epsilon$-neighborhood of $x$ is defined as $B_\epsilon \left({x}\right) \setminus \left\{{x}\right\}$.

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


It can also be defined as:

$\left\{{y \in A: 0 < d \left({x, y}\right) < \epsilon}\right\}$


Topology

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x \in S$.

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


Then $V \setminus \left\{{x}\right\}$ is called a deleted neighborhood of $x$.

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


Also known as

Also called a punctured neighborhood.


Sources