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.

Examples

$1$-Neighborhood of $2$

The $1$-neighborhood of $2$ is the set:

$\map {N_1} 2 = \openint 1 3 = \set {x \in \R: \size {x - 2} < 1}$

Linguistic Note

The UK English spelling of neighborhood is neighbourhood.

Sources

• 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 11$. Continuity on the Euclidean line