Definition:Open Set/Real Analysis/Real Numbers

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

Then $I$ is open (in $\R$) :
 * $\forall x_0 \in I: \exists \epsilon \in \R_{>0}: \left({x_0 - \epsilon\,.\,.\,x_0 + \epsilon}\right) \subseteq I$

where $\left({x_0 - \epsilon\,.\,.\,x_0 + \epsilon}\right)$ is an open interval.

Note that $\epsilon$ may depend on $x_0$.

Also see

 * Definition:Closed Subset of Real Numbers