Definition:Open Set/Real Analysis

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

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

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

Then $I$ is described as open (in $\R$).

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

By Open Sets in Real Number Line every open set $I \subseteq \R$ is a countable union of pairwise disjoint open intervals:


 * $\displaystyle I = \bigcup_{n \mathop \in \N} \left({a_n \,.\,.\, b_n}\right) \subseteq \R$