Definition:Isolated Point (Real Analysis)

Definition

Let $S \subseteq \R$ be a subset of the set of real numbers.

Let $\alpha \in S$.

Then $\alpha$ is an isolated point of $S$ iff there exists an open interval of $\R$ whose midpoint is $\alpha$ which contains no points of $S$ except $\alpha$:

$\exists \epsilon \in \R, \epsilon > 0: \left({\alpha - \epsilon .. \alpha + \epsilon}\right) \cap S = \left\{{\alpha}\right\}$

By definition of neighborhood in the context of the real number line under the usual (Euclidean) metric:

$N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon .. \alpha + \epsilon}\right)$

it can be seen that this definition is compatible with that for a metric space:

$\exists \epsilon \in \R, \epsilon > 0: N_\epsilon \left({\alpha}\right) \cap S = \left\{{\alpha}\right\}$