# Diameters of Sets

1. Oct 11, 2011

### Kindayr

1. The problem statement, all variables and given/known data
Find a condition on a metric space $(X,d)$ that ensures that there exist subsets $A,B$ of $X$ with $A\subset B$ such that $diam(A)=diam(B)$.

2. Relevant equations
$diam(A)=\sup\{d(r,s):r,s\in A\}$;
$A\subseteq B\implies diam(A)\leq diam(B)$.

3. The attempt at a solution
Well I know examples of where this is true (ie, let $A=(-\infty,5]\cup [-5,\infty)\subset (-\infty,4]\cup [-4,\infty)=B$). But I don't know which condition allows this to be true. Any help is good. Thank you!

2. Oct 12, 2011

### Kindayr

Bump. Anyone? Any Idea?