Definition:Open Set/Real Analysis

From ProofWiki
Jump to: navigation, search

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:

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

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


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


Real Euclidean Space

Let $n\geq1$ be a natural number.

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


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

$\forall x\in U : \exists R>0 : B(x, R) \subset U$

where $B(x,R)$ is the open ball of radius $R$ centered at $x$.


Also see

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