\begin{definition}[Definition:Deleted 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:
:$\set {y \in A: 0 < \map d {x, y} < \epsilon}$
\end{definition}