Definition:Deleted Neighborhood/Normed Vector Space

Definition

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:

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

Also known as

A deleted neighborhood is also called a punctured neighborhood.

Also see

These definitions are seen to be equivalent by the definition of open $\epsilon$-ball neighborhood.