Definition:Open Set/Real Analysis/Real Euclidean Space

Definition

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:

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

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

Also see

• Definition:Closed Subset of Real Euclidean Space
• Definition:Neighborhood in Real Euclidean Space