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 $S$ if and only if there exists an open interval of $\R$ whose midpoint is $\alpha$ which contains no points of $S$ except $\alpha$:

$\exists \epsilon \in \R_{>0}: \openint {\alpha - \epsilon} {\alpha + \epsilon} \cap S = \set \alpha$

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

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

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

$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} \alpha \cap S = \set \alpha$