# Definition:Neighborhood (Metric Space)

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*This page is about Neighborhood in the context of Metric Space. 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 $M$**.

### Neighborhood of Compact Subset

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:

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

## 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**.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Definition $4.4$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**neighbourhood** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**neighbourhood**:**1.**