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:
- $\forall x_0 \in I: \exists \epsilon \in \R_{>0}: \openint {x_0 - \epsilon} {x_0 + \epsilon} \subseteq I$
where $\openint {x_0 - \epsilon} {x_0 + \epsilon}$ is an open interval.
Note that $\epsilon$ may depend on $x_0$.
Also see
Sources
- 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 11$. Continuity on the Euclidean line