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 $\epsilon \in \R: \epsilon > 0$ is a positive real number.

From the definition of the real numbers as a metric space, it can be seen that this definition is compatible with the definition of a open $\epsilon$-ball neighborhood in a metric space.

Linguistic Note
The UK English spelling of this is neighbourhood.