Definition:Deleted Neighborhood/Metric Space

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

Let $x \in A$.

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

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

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

It can also be defined as:

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

Also see

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