Definition:Neighborhood (Metric Space)

Definition
Let $M = \struct {A, d}$ 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:
 * $\map {B_\epsilon} x \subseteq S$

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

Also known as
A neighborhood of $x$ that has been created around an open $\epsilon$-ball at $x$ is sometimes referred to as an $\epsilon$-neighborhood of $x$.

Also see

 * Definition:Open Ball of Metric Space
 * Definition:Open Set of Metric Space