Definition:Neighborhood (Metric Space)

This page is about neighborhood in the context of metric spaces. For other uses, see neighborhood.

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 $M$.

Let $M = \struct {A, d}$ be a metric space.

Let $K \subseteq A$ be a compact subset of $A$.

The $\epsilon$-neighborhood of $K$ in $M$ defined and denoted as:

$\map {\NN_\epsilon} K := \set {x \in A: \exists y \in K: \map d {x, y} \le \epsilon}$

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

The UK English spelling of neighborhood is neighbourhood.