Diameter of open and closed balls

Question

I'm homelearning calculus and metric spaces.

We are in a space with euclidean metric. In my book, it is stated that the diameter of $S(1,2)=2$ and the diameter of $\bar{S}(1,2)=4$, where $S$ is an open ball and $\bar{S}$ is a closed ball.

Could you please explain to me why are the diameters $2$ and $4$? I'm not sure how they got that numbers, I know the definitions but I don't quite understand them.

Thanks

What is the definition of diameter you are using ? – Someone Apr 03 '20 at 13:47 — Someone, Apr 03 '20 at 13:47
@Something The diamater of $D(A)$ is the supremum $D(A)$. – LukasT Apr 03 '20 at 13:49 — LukasT, Apr 03 '20 at 13:49

score 3 · Accepted Answer · answered Apr 03 '20 at 13:52

Well it depends on where you're working.

For example, if the space you're working on is $\mathbb R$, then $S(1,2)=(-1,3)$ and $\bar{S}(1,2)=[-1,3]$, and thus the diameter of both sets is 4, while if you're working on $\mathbb Z$, then $S(1,2)=\{0,1,2\}$ has diameter $2$ and $\bar{S}(1,2)=\{-1,0,1,2,3\}$ has diameter $4$.

Diameter of open and closed balls

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