Definition:Neighborhood (Real Analysis)/Epsilon
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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