Definition:Open Set/Real Analysis/Real Numbers

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Definition

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


Then $I$ is open (in $\R$) if and only if:

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

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


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


Also see