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

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

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Balls of Subspaces

Let $(X,d)$ be a metric space then by $Y\subseteq X$ is a subspace then

$$B_Y(y,r)=\{z\in Y:d(z,y)<r\}$$

is for the open ball about $y$ of radius $r$ in $Y$

With notable property

$$B_X(y,r)=\{x\in X:d(x,y)<r\}$$

is for the open ball for radius $r$ about $y$ in $X$ with property then

$$B_Y(y,r)=Y\cap B_X(y,r)$$

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Totally Bounded

Let $X$ be a metric space then

$X$ is totally bounded if, for any $ϵ>0$ then

$$\text{it may be covered by finitely many open balls of radius }ϵ$$