Open Ball

Topics: Calculus - Topology

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(definition)

We define the open ball of radius $r\in \mathbb{R}$ around $x\in {\mathbb{R}}^n$ as:

$$B_r(x)=\{u\in {\mathbb{R}}^n\mid \Vert x-u\Vert <r\}$$

Examples

Take the following ball:

$$B_1(0)\subseteq \mathbb{R}=\{u\in \mathbb{R}\mid \Vert u-0\Vert <1\}=(-1,1)$$

This is a pretty easy to visualise ball that exists in $\mathbb{R}$, since it’s just an open interval.

Now take this other ball:

$$B_1(0,0)=\{(x,y)\in {\mathbb{R}}^2\mid \Vert (x,y)-(0,0)\Vert <1\}=\sqrt{x^2+y^2}<1$$

This ball is an open circle in ${\mathbb{R}}^2$ which is also rather easy to visualise. An analogue ball in ${\mathbb{R}}^3$ would be an open sphere.

Open balls are used for defining interior points, limit points and the limit of a sequence.