9.1 Set Properties

Note

A set $\mathcal{X}\subseteq {\mathbb{R}}^n$ is open if for any $x\in \mathcal{X}$, there exists an Euclidean ball $B_{ϵ}(x)=\{y\in {\mathbb{R}}^n:\Vert y-x{\Vert }_2\}$ with some radius $ϵ>0$, such that $B_{ϵ}(x)\subseteq \mathcal{X}$. Similarly, $\mathcal{X}$ is said to be closed if its complement ${\mathbb{R}}^n\backslash \mathcal{X}$ is open. An example of an open set is any interval $(a,b)$.