What is the ball associated with a certain metric?

Question

I do not understand what the ball associated with a certain metric is. How do you obtain the radius and centre of such ball? Should you just take an arbitrary point?

Answer 1

For a metric $d$ on a set $S$ and for $p\in S$ and for $r\in \Bbb R$ the $open$ ball $B_d(p,r)$ is defined as $$B_d(p,r)=\{q\in S: d(q,p)<r\}.$$ Non-positive values of $r$ are rarely mentioned because $B_d(p,r)=\emptyset$ when $r\leq 0.$

When only one metric $d$ is under discussion, $B_d(p,r)$ is often written as $B(p,r).$

Some people write $B_r(p)$ instead, which I think is harder to read, and can become awkward when $r$ is defined by a formula instead of a single letter.

$p$ is called a center of $B_d(p,r)$ and $r$ is called a radius of $B_d(p,r).$ (Note: Not necessarily "the" center nor "the" radius. See below.)

In some metric spaces (...e.g. $S=\Bbb R$ with $d(p,q)=|p-q|...),$ if $r,r'>0$ and $B(p,r)=B(p',r')$ then $p=p'$ and $r=r'.$ But in general this may not happen.

The simplest example is to let $S$ have more than one member, and let $d(p,p')=1$ when $p,p'$ are unequal members of $S$. (This is called the discrete metric on $S.$) Then for any $p,p'\in S$ we have $B_d(p,2)=B_d(p',3)=S. $ And for any $r,r'\in (0,1]$ and any $p\in S$ we have $B_d(p,r)=B_d(p,r')=\{p\}.$