Definition:Deleted Neighborhood

Topology
Let $x \in X$ be a point in a topological space with topology $\vartheta$.

Let $V \subseteq X$ be a neighborhood of $x$.

Then a deleted neighborhood of $x$ is $V - \left\{{x}\right\}$.

That is, it is a neighborhood of $x$ with $x$ itself removed.

Metric Space
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 $N_\epsilon \left({x}\right) - \left\{{x}\right\}$.

It can also be defined as $\left\{{y \in A: 0 < d \left({x, y}\right) < \epsilon}\right\}$.

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