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.