Possible forms of open balls

Question

Consider $X= ( \Bbb Q \cap [ 0,3] , d_E)$The question is as such:

"Describe the possible forms that an open ball can take in $X = (\Bbb Q ∩ [0, 3], d_E )$."

I don't understand this means exactly. Can anyone shed some light on this matter?

Answer 1

The open unit ball around $0$ in the Euclidean metric is $B_1(0)=\{x\in X:|x|<1\}$. Any open ball in that metric is $B_r(y)=\{x\in X:|x-y|<r\}$.

The shape of a ball is either disjoint points on the the real line which are dense in some interval if the ball is disjoint from $[0,3]$, a whole interval if the ball is contained in $[0,3]$, or a mix of those when the ball intersects with $[0,3]$ but is not contained in it.