# Definition:Neighborhood (Metric Space)

*This page is about neighborhoods in the context of metric spaces. For other uses, see Definition:Neighborhood.*

## Contents

## 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 $M$**.

## Also see

- Results about
**neighborhoods**can be found here.

## Linguistic Note

The UK English spelling of **neighborhood** is **neighbourhood**.

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.4$: Open Balls and Neighborhoods: Definition $4.4$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: Compactness