# Question regarding diameter of subsets of a metric space

The question is :

Find a condition on a metric space$(X,d)$ that ensures that there exist subsets $A$ and $B$ of $X$ with $A \subset B$ such that $diam(A)$ = $diam(B)$.

I know that if $X$ is a metric space and $A$ and $B$ are subsets of $X$ with $A \subset B$ then

$diam(A)$ <= $diam(B)$.

If i assume that the metric $d$ on $X$ is the discrete metric then then diameters of $A$ and $B$ will be the same.

Is this good enough ? or can there be some other condition on the metric space $X$ ?

• HINT: >A: [0, 1] >B: [-1, 0, 1]. Yes you only need the discrete metric. Why are you trying to overcomplicate it? – Don Larynx Oct 9 '13 at 11:31