Definition:Deleted Neighborhood/Metric Space

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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:

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

Also see

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