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 $x \in S$.

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

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

Linguistic Note
The UK English spelling of this is neighbourhood.

Also see

 * Definition:Open Ball
 * Definition:Open Set (Metric Space)