Hi there. I was working on this problem. And I wanted to know if my demonstration is right, I'm not sure if it's complete.

The problem says: The distance D(A,B) between two nonempty subsets A and B of a metric space (X,d) is defined to be:

$\displaystyle D(A,B)=inf d(a,b),a\in{A},b\in{B}$

Inf denotes the infimus.

Show that D does not define a metric on the power set of X.

Well, so what I considered is that if a belongs to A, and b belongs to B, but a doesn't belongs to B, and b doesn't belongs to A (i.e. A and B has no elements in common), then the distance could never be zero. But I don't know if this is enough to show that it doesn't define a metric.

Anyway, I could show using the sets $\displaystyle A={0,1,2,3...}, B={-1,1,-1,...,(-1)^n},C={0,-1,-2,...}$

That the triangle inequality insn't accomplished, so I have a counter example.

Bye there, and thanks in advance.