Understanding closed ball in the discrete metric system

Question

Let $(X,d)$ be a discrete metric space and $x \in X$. Describe the open ball $B(x,1)$ and the closed ball $B[x,r]$.

I understand the first ball. If distance is between $0$ and $1$ then we say that $y$ is any element of ball then $x=y$ .

I don't understand the second ball. When we think the same as like open ball ,then $x=y$ but it also can be $1$ so this cause $X$(the whole set).How to select which one is answer ? And how can it be whole set, is this whole set is just center which is radius 1 or different thing?

how to write this part , ı dont know what I can write .maybe set of solution is okey? — 따라 오세요, Mar 31 '23 at 08:56
Does $B[x,r]$ denote the closed ball of radius $r$ centered at $x$? — luxerhia, Mar 31 '23 at 09:09
You may divide into cases according to whether $0 \leq r < 1$ or $r \geq 1$. — luxerhia, Mar 31 '23 at 09:17
I understand 0≤r<1, the solution is only x=y becasue distence can be either 0 or 1 and in this way distance is 0 so x=y — 따라 오세요, Mar 31 '23 at 09:18
But I dont understand r≤1 this situation because it include both whole set and only point like in the previous .Which one do ı take as an answer — 따라 오세요, Mar 31 '23 at 09:20
Also, if answer is a whole set , which means that is only circle with radius one or different thing. — 따라 오세요, Mar 31 '23 at 09:22
In a matric space $(M,d)$, an open ball is a set $$B(x,r)={y\in M : d(x,y)<r}$$
And a closed ball is a set $$\overline{B}(x,r)={y\in M : d(x,y)\leq r}$$

Split into four cases:

$r=0$,

$0<r<1$,

$r=1$,

$r>1$

And think about how the open and closed balls will differ. — K.defaoite, Mar 31 '23 at 10:12

score 0 · Accepted Answer · edited Mar 31 '23 at 13:31

0

For $0<r<1$ we have $B(x,r)=\{x\}$ and $B[x,r]=\{x\}$. Now for $r=1$ we have $B(x,r)=\{x\}$ and $B=[x,r]=X$. And finally for $r>1$ we have $B(x,r)=B[x,r]=X$.

edited Mar 31 '23 at 13:31

gist076923

1,266

answered Mar 31 '23 at 13:21

Julian A. A.

26

Understanding closed ball in the discrete metric system

1 Answers1