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:


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

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.