Definition:Deleted Neighborhood (Metric Space)

Definition

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

Let $x \in A$.

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

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

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

It can also be defined as:

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

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