Definition:Neighborhood (Metric Space)

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

Let $S \subseteq A$ be a subset of $A$.

Let $\exists \epsilon \in \R_{>0}$ such that the $\epsilon$-ball at $x$ lies completely in $S$, that is:
 * $B \left({x; \epsilon}\right) \subseteq S$

Then $S$ is a neighborhood of $x$.

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

Let $S \subseteq A$ be a subset of $A$.

Then $S$ is an open neighborhood (of $M$) iff it is a neighborhood of each of its points.

Linguistic Note
The UK English spelling of this is neighbourhood.