Definition:Neighborhood (Real Analysis)/Epsilon

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 this is neighbourhood.