\begin{definition}[Definition:Isolated Point (Real Analysis)]
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 $\alpha$:
:$\exists \epsilon \in \R_{>0}: \openint {\alpha - \epsilon} {\alpha + \epsilon} \cap S = \set \alpha$
{{ExtractTheorem|Add the below as a separate proof page}}
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:
:$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} \alpha \cap S = \set \alpha$
{{expand|Isolated Point of Equation}}
Category:Definitions/Real Analysis
\end{definition}