Definition:Open Set/Real Analysis

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Definition

Real Numbers

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 $x_0$.


Real Euclidean Space

Let $n \ge 1$ be a natural number.

Let $U \subseteq \R^n$ be a subset.


Then $U$ is open (in $\R^n$) if and only if:

$\forall x \in U : \exists R > 0: \map B {x, R} \subset U$

where $\map B {x, R}$ denotes the open ball of radius $R$ centered at $x$.


Also see

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