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

• Results about neighborhoods can be found here.

Linguistic Note

The UK English spelling of neighborhood is neighbourhood.