Definition:Deleted Neighborhood/Normed Vector Space
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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.