# Definition:Neighborhood (Real Analysis)

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*This page is about neighborhoods in the real number line. For other uses, see Definition:Neighborhood.*

## Definition

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

### Open Subset Neighborhood

Let $N_\alpha$ be a subset of $\R$ which contains (as a subset) an open real set which itself contains (as an element) $\alpha$.

Then $N_\alpha$ is a **neighborhood** of $\alpha$.

### Epsilon-Neighborhood

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.

## Linguistic Note

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