# Definition:Neighborhood (Real Analysis)/Epsilon

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## Contents

## Definition

Let $\alpha \in \R$ be a real number.

On the real number line with the usual metric, the **$\epsilon$-neighborhood** of $\alpha$ is defined as the open interval:

- $\map {N_\epsilon} \alpha := \openint {\alpha - \epsilon} {\alpha + \epsilon}$

where $\epsilon \in \R_{>0}$ is a (strictly) positive real number.

## Also presented as

The **$\epsilon$-neighborhood** of $\alpha$ can also be presented as:

- $\map {N_\epsilon} \alpha := \set {x \in \R: \size {x - \alpha} < \epsilon}$

## Also see

- Real Number Line is Metric Space: this definition is compatible with that of an open $\epsilon$-ball neighborhood in a metric space.

## Examples

### $1$-Neighborhood of $2$

The $1$-neighborhood of $2$ is the set:

- $\map {N_1} 2 = \openint 1 3 = \set {x \in \R: \size {x - 2} < 1}$

## Linguistic Note

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

## Sources

- 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 11$. Continuity on the Euclidean line