# Definition:Neighborhood (Real Analysis)/Epsilon

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## 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:

- $N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon \,.\,.\, \alpha + \epsilon}\right)$

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

## Also see

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

## Linguistic Note

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