09 - Balls

Open Ball

Let $X$ be a metric space

If $a\in X$ and $ϵ>0$ then the open ball of radius $ϵ$ about $\alpha$ is the set

$$B(a,ϵ)=\{x\in X:d(x,a)<ϵ\}$$

Closed Ball

Let $X$ be a metric space

If $a\in X$ and $ϵ>0$ then the closed ball of radius $ϵ$ about $\alpha$ is the set

$$\underline{B}(a,ϵ)=\{x\in X:d(x,a)\le ϵ\}$$